├── tareas ├── midterm │ ├── Makefile │ └── main.tex ├── projects │ ├── Makefile │ └── main.tex ├── tarea1 │ ├── Makefile │ └── main.tex ├── tarea2 │ ├── Makefile │ └── main.tex ├── tarea3 │ ├── Makefile │ └── main.tex ├── tarea4 │ ├── Makefile │ └── main.tex ├── tarea5 │ ├── Makefile │ └── main.tex ├── tarea-topicos-1 │ ├── Makefile │ └── main.tex └── tarea-topicos-2 │ ├── Makefile │ └── main.tex ├── Makefile ├── convergence_rates_FD.csv ├── README.md ├── convergence_rates_FD.py ├── nl-poisson.py ├── main.tex ├── wave.py ├── optimization-poisson.py ├── chapters ├── introduction.tex ├── beyond-linearity.tex ├── time-dependent.tex └── weak-forms-galerkin.tex ├── .gitignore ├── src └── macros.tex └── main.bib /tareas/midterm/Makefile: -------------------------------------------------------------------------------- 1 | all: 2 | pdflatex main.tex 3 | 4 | -------------------------------------------------------------------------------- /tareas/projects/Makefile: -------------------------------------------------------------------------------- 1 | all: 2 | pdflatex main.tex 3 | 4 | -------------------------------------------------------------------------------- /tareas/tarea1/Makefile: -------------------------------------------------------------------------------- 1 | all: 2 | pdflatex main.tex 3 | 4 | -------------------------------------------------------------------------------- /tareas/tarea2/Makefile: -------------------------------------------------------------------------------- 1 | all: 2 | pdflatex main.tex 3 | 4 | -------------------------------------------------------------------------------- /tareas/tarea3/Makefile: -------------------------------------------------------------------------------- 1 | all: 2 | pdflatex main.tex 3 | 4 | -------------------------------------------------------------------------------- /tareas/tarea4/Makefile: -------------------------------------------------------------------------------- 1 | all: 2 | pdflatex main.tex 3 | 4 | -------------------------------------------------------------------------------- /tareas/tarea5/Makefile: -------------------------------------------------------------------------------- 1 | all: 2 | pdflatex main.tex 3 | 4 | -------------------------------------------------------------------------------- /tareas/tarea-topicos-1/Makefile: -------------------------------------------------------------------------------- 1 | all: 2 | pdflatex main.tex 3 | 4 | -------------------------------------------------------------------------------- /tareas/tarea-topicos-2/Makefile: -------------------------------------------------------------------------------- 1 | all: 2 | pdflatex main.tex 3 | 4 | -------------------------------------------------------------------------------- /Makefile: -------------------------------------------------------------------------------- 1 | all: 2 | pdflatex main.tex 3 | bibtex main.aux 4 | pdflatex main.tex 5 | pdflatex main.tex 6 | 7 | -------------------------------------------------------------------------------- /convergence_rates_FD.csv: -------------------------------------------------------------------------------- 1 | h,fd,bd,cd 2 | 0.1111111111111111,2.1601048215852914,2.160104821585292,0.468057906191615 3 | 0.030303030303030304,0.5974806053596746,0.5974806053596782,0.03772265521841689 4 | 0.008547008547008548,0.16869598221691198,0.16869598221691462,0.003018543420597375 5 | 0.0024752475247524753,0.04885844289959289,0.048858442899591355,0.00025329135780083334 6 | 0.0007204610951008645,0.014221307704910485,0.014221307704907551,2.1458975227872656e-05 7 | 0.00020964360587002095,0.004138198611140266,0.0041381986111345415,1.8169862174843843e-06 8 | 6.105751618024179e-05,0.0012052270534362842,0.00120522705343335,1.541277461214463e-07 9 | 1.778315224156634e-05,0.0003510253564070237,0.00035102535640687974,1.3091789874408732e-08 10 | 5.179495413556811e-06,0.00010223915154666027,0.00010223915154512135,1.2020802131473829e-09 11 | 1.5085937040350355e-06,2.977850681835535e-05,2.9778506816816215e-05,4.254472329989767e-10 12 | 4.393972348732009e-07,8.673594438239791e-06,8.673594435306374e-06,1.1958620760310623e-09 13 | 1.279802439457026e-07,2.5270792628068195e-06,2.5270792620374696e-06,5.159537685983651e-09 14 | 3.727593852705694e-08,7.388548984924026e-07,7.388896956228574e-07,1.3876971571846752e-08 15 | 1.0857111284663241e-08,2.289682039524621e-07,2.29066990042881e-07,4.4329122594888304e-08 16 | 3.1622776703367586e-09,2.917237234001391e-07,2.920527819583185e-07,1.2349160005697968e-07 17 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Context 2 | 3 | These notes were born as a way to keep track of the contents I intended to show in a course on Advanced Topics for Mathematical Engineering at Pontificia Universidad Católica de Chile, the second semester of 2024. The idea of the course is to give a modern set of tools for a rigorous computational scientist, so that they can understand the complexity of a given problem and squeeze the most mathematics out of a problem whenever possible. This should help in developing efficient numerical methods and robust strategies to be able to trust the computational pipelines that they propose. 4 | 5 | # Contributing 6 | 7 | For contributions, please follow the format of the manuscript, which should be fairly common mathematical notation. 8 | 9 | - Create vectors, matrices, and tensors with `\vec`, `\mat`, and `\ten`. 10 | - Vector norms (and semi-norms) are $|\cdot|$. Function and operator norms are $\lVert \cdot \rVert$. 11 | - Duality pairings and inner products are $\langle \cdot, \cdot\rangle$ 12 | - Small sections can be introduced with `\paragraph` instead of a `\subsubsubsubsubsubsubsubsubsubsection`. 13 | 14 | Contributors may be added as editors upon request. Otherwise, pull requests are both encouraged and desired. Thanks in advance for the help in making these notes clear and useful :) . 15 | 16 | # Disclaimer 17 | 18 | I have been writing most things in a rush, so help in finding typos and mistakes will be much appreciated. Raise it either by email (nicolas.barnafi at uc.cl) or directly here as an issue. Using git for everything is a personal pathology. 19 | 20 | -------------------------------------------------------------------------------- /convergence_rates_FD.py: -------------------------------------------------------------------------------- 1 | import numpy as np 2 | import matplotlib.pyplot as plt 3 | import time 4 | import pandas as pd 5 | 6 | Ns = [round(10**exp) for exp in np.linspace(1,8.5,15)] 7 | 8 | fig, ax = plt.subplots(figsize=(3,3), dpi=150) 9 | 10 | fderr = [] 11 | bderr = [] 12 | cderr = [] 13 | hs = [] 14 | 15 | for N in Ns: 16 | xs = np.linspace(0, 1, N) 17 | h = xs[1] - xs[0] 18 | hs.append(h) 19 | ys = np.sin(2*np.pi*xs) 20 | true_deriv = 2*np.pi*np.cos(2*np.pi*xs) 21 | 22 | # forward difference 23 | 24 | fd = [] 25 | bd = [] 26 | cd = [] 27 | 28 | t0 = time.time() 29 | 30 | # Forward difference (excluding last point) 31 | fd = (ys[1:] - ys[:-1]) / h 32 | 33 | # Backward difference (excluding first point) 34 | bd = (ys[1:] - ys[:-1]) / h 35 | 36 | # Central difference (excluding first and last points) 37 | cd = (ys[2:] - ys[:-2]) / (2 * h) 38 | 39 | fderr.append(np.max(np.abs(fd - true_deriv[:-1]))) 40 | bderr.append(np.max(np.abs(bd - true_deriv[1:]))) 41 | cderr.append(np.max(np.abs(cd - true_deriv[1:-1]))) 42 | print(f"done with N = {N} in {time.time() - t0} s.") 43 | 44 | ax.loglog(hs, fderr, label='forward') 45 | ax.loglog(hs, bderr, '--', label='backward') 46 | ax.loglog(hs, cderr, label='centered') 47 | ax.set_ylabel("error sup norm") 48 | ax.set_xlabel("h") 49 | 50 | plt.legend() 51 | plt.tight_layout() 52 | plt.show() 53 | 54 | results = { 55 | 'h': hs, 56 | 'fd': fderr, 57 | 'bd': bderr, 58 | 'cd': cderr 59 | } 60 | 61 | pd.DataFrame(results).to_csv('convergence_rates_FD.csv', index=False) 62 | -------------------------------------------------------------------------------- /nl-poisson.py: -------------------------------------------------------------------------------- 1 | from firedrake import * 2 | import matplotlib.pyplot as plt 3 | 4 | N = 20 5 | p = 10 # Exponent, 5 nice 6 | F = 1 # 20 nice 7 | u0 = 1 # 100.0 nice 8 | 9 | f = Constant(F) 10 | mesh = UnitSquareMesh(N, N) 11 | V = FunctionSpace(mesh, 'CG', 1) 12 | bcs = DirichletBC(V, Constant(0), "on_boundary") 13 | 14 | u = Function(V) 15 | v = TestFunction(V) 16 | 17 | 18 | params = {"snes_converged_reason": None, "ksp_type": "preonly", "pc_type": "lu"} 19 | def solver(formulation): 20 | 21 | if formulation == "newton": 22 | u.interpolate(Constant(u0)) # Restart from the previously computed solution... 23 | F = dot(grad(u), grad(v)) * dx + (u**p - f) * v * dx 24 | solve(F == 0, u, bcs=bcs, solver_parameters=params) 25 | 26 | else: # picard 27 | 28 | u.interpolate(Constant(u0)) 29 | w = Function(V) 30 | w.interpolate(Constant(u0)) 31 | F = dot(grad(u), grad(v)) * dx + (pow(u,p) - f) * v * dx 32 | Fw = dot(grad(u), grad(v)) * dx + (pow(w,p-1)*u - f) * v * dx 33 | 34 | res_vec = assemble(F, bcs=bcs) 35 | err = sqrt(res_vec.vector().inner(res_vec.vector())) 36 | it = 0 37 | err0 = err 38 | tol = 1e-6 39 | maxit = 100 40 | print(f"It {it}, err={err} ") 41 | 42 | while err / err0 > tol and it < maxit: 43 | solve( Fw == 0, u, bcs=bcs, solver_parameters=params ) 44 | assemble(F, bcs=bcs, tensor=res_vec) 45 | err = sqrt(res_vec.vector().inner(res_vec.vector())) 46 | w.assign(u) 47 | print(f"It {it}, err={err}") 48 | it += 1 49 | 50 | print("==================== Newton") 51 | #solver("newton") 52 | print("==================== Fixed-point") 53 | solver("picard") 54 | File("output/nl.pvd").write(u) 55 | 56 | 57 | 58 | 59 | -------------------------------------------------------------------------------- /main.tex: -------------------------------------------------------------------------------- 1 | \documentclass{scrbook} 2 | \input{src/macros.tex} 3 | 4 | \title{Applications of Functional Analysis and PDEs} 5 | \author{Nicol\'as A Barnafi} 6 | \date{} % Keeping for 'last update' track 7 | 8 | \begin{document} 9 | 10 | \maketitle 11 | \tableofcontents 12 | \mainmatter 13 | 14 | \input{chapters/introduction.tex} 15 | 16 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 17 | \chapter{Preliminaries}\label{chapter:preliminaries} 18 | \input{chapters/preliminaries.tex} 19 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 20 | \chapter{Finite differences}\label{chapter:finite-differences} 21 | \input{chapters/finite-differences.tex} 22 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 23 | \chapter{Sobolev spaces}\label{chapter:sobolev} 24 | \input{chapters/sobolev-spaces.tex} 25 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 26 | \chapter{Weak formulations and Galerkin methods}\label{chapter:weak-forms-galerkin} 27 | \input{chapters/weak-forms-galerkin.tex} 28 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 29 | \chapter{Finite elements}\label{chapter:fem} 30 | \input{chapters/fem.tex} 31 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 32 | \chapter{Sturm-Liouville theory}\label{chapter:sturm-liouville} 33 | \input{chapters/sturm-liouville.tex} 34 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 35 | \chapter{Beyond ellipticity}\label{chapter:beyond-ellipticity} 36 | \input{chapters/beyond-ellipticity.tex} 37 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 38 | \chapter{Beyond linearity}\label{chapter:nonlinear} 39 | \input{chapters/beyond-linearity.tex} 40 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 41 | \chapter{Time dependent problems}\label{chapter:dependent} 42 | \input{chapters/time-dependent.tex} 43 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 44 | \chapter{Continuum mechanics}\label{chapter:continuum} 45 | \input{chapters/continuum-mechanics.tex} 46 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 47 | \bibliography{main} 48 | \bibliographystyle{alpha} 49 | \end{document} 50 | 51 | -------------------------------------------------------------------------------- /wave.py: -------------------------------------------------------------------------------- 1 | from firedrake import * 2 | 3 | EXPLICIT=0 4 | IMPLICIT=1 5 | SYMPLECTICU=2 6 | SYMPLECTICV=3 7 | 8 | 9 | N = 50 10 | P = 2 11 | mesh = UnitSquareMesh(N, N) 12 | time_scheme = SYMPLECTICV 13 | 14 | mu = Constant(1e-1, domain=mesh) 15 | dt = 1e-2 16 | tf = 20 17 | save_every = 10 18 | 19 | V = FunctionSpace(mesh, "CG", P) 20 | W = V * V 21 | h = Function(V, name="h") 22 | du = TrialFunction(V) 23 | dv = TestFunction(V) 24 | v, u = TrialFunctions(W) 25 | vt, ut = TestFunctions(W) 26 | sol = Function(W) 27 | un = Function(V, name="u") 28 | vn = Function(V) 29 | 30 | idt = Constant(1/dt) 31 | H = 0.5 * mu * (grad(sol.sub(1)))**2 + 0.5 * sol.sub(0)**2 32 | if time_scheme == EXPLICIT: 33 | us = un 34 | vs = vn 35 | elif time_scheme == IMPLICIT: 36 | us = u 37 | vs = v 38 | elif time_scheme == SYMPLECTICU: 39 | us = u 40 | vs = vn 41 | else: # SYMPLECTICV 42 | us = un 43 | vs = v 44 | 45 | a = idt * (v - vn) * vt * dx + idt * (u - un) * ut * dx - \ 46 | vs * ut * dx + mu * inner(grad(us), grad(vt)) * dx 47 | 48 | X = mesh.coordinates 49 | C = Constant((0.5, 0.5)) 50 | XC = X-C 51 | r = sqrt(dot(XC, XC)) 52 | u0 = conditional(lt(r, 0.3), Constant(1), 0) 53 | solve(Constant(1) * du*dv*dx + Constant(0.5) * CellDiameter(mesh)**2 * dot(grad(du), grad(dv))*dx==u0*dv*dx, un, bcs=DirichletBC(V, u0, "on_boundary")) 54 | #un.interpolate(u0) 55 | sol.sub(1).assign(un) 56 | sol.sub(0).interpolate(Constant(0)) 57 | 58 | problem = LinearVariationalProblem(lhs(a), rhs(a), sol, constant_jacobian=True) 59 | 60 | params = {"ksp_type": "gmres", 61 | "mat_type": "nest", 62 | "ksp_norm_type": "unpreconditioned", 63 | "ksp_atol": 0.0, 64 | "ksp_rtol": 1e-6, 65 | "pc_type": "fieldsplit", 66 | "pc_fieldsplit_type": "multiplicative", 67 | "fieldsplit_0_ksp_type": "preonly", 68 | "fieldsplit_0_pc_type": "jacobi", 69 | "fieldsplit_1_ksp_type": "preonly", 70 | "fieldsplit_1_pc_type": "hypre" 71 | } 72 | 73 | solver = LinearVariationalSolver(problem, solver_parameters=params) 74 | 75 | outfile = File("output/wave.pvd") 76 | t = 0 77 | h.interpolate(H) 78 | outfile.write(un, h, t=t) 79 | i = 0 80 | energies = [] 81 | while t < tf: 82 | if i % 10 == 0: print("Solving t={:.2f}".format(t)) 83 | solver.solve() 84 | vn.assign(sol.sub(0)) 85 | un.assign(sol.sub(1)) 86 | t += dt 87 | if i % save_every == 0: 88 | h.interpolate(H) 89 | outfile.write(un, h, t=t) 90 | energies.append(assemble(H * dx)) 91 | i += 1 92 | import matplotlib.pyplot as plt 93 | plt.plot(energies) 94 | plt.show() 95 | -------------------------------------------------------------------------------- /optimization-poisson.py: -------------------------------------------------------------------------------- 1 | from firedrake import * 2 | import matplotlib.pyplot as plt 3 | import scipy.sparse as sp 4 | 5 | alpha = 1e-1 6 | rtol = 1e-8 7 | maxit = 1000 8 | verbose = False 9 | 10 | # Poisson 11 | def solvePoisson(N, grad_type, alpha=0.1, verbose=False): 12 | mesh = UnitSquareMesh(N, N) 13 | V = FunctionSpace(mesh, 'CG', 1) 14 | 15 | f = Constant(1.0) 16 | u = TrialFunction(V) 17 | v = TestFunction(V) 18 | 19 | bc = DirichletBC(V, Constant(0), "on_boundary") 20 | 21 | sol = Function(V) 22 | grad_sol = Function(V) 23 | 24 | err=1.0 25 | err_vec = assemble( dot(grad(sol), grad(v)) * dx - f * v * dx, bcs=bc) 26 | with err_vec.dat.vec_ro as vv: 27 | err0 = vv.norm() 28 | 29 | it = 0 30 | if verbose: 31 | print(f"It {it:4}, error={err:2.2e}") 32 | while err > rtol and it < maxit: 33 | 34 | if grad_type == "L2": 35 | inner = lambda _u: _u*v*dx 36 | elif grad_type == "H01": 37 | inner = lambda _u: dot(grad(_u), grad(v)) * dx 38 | elif grad_type == "H1": 39 | inner = lambda _u: (_u*v + dot(grad(_u), grad(v)) ) * dx 40 | else: # l2 41 | L = dot(grad(sol), grad(v)) * dx - f * v * dx 42 | 43 | # Extract a sparse python matrix 44 | res = assemble(L, bcs=bc) 45 | sol.vector().axpy(-alpha, res) 46 | 47 | if grad_type != "l2": 48 | a = inner(u) 49 | L = dot(grad(sol), grad(v)) * dx - f * v * dx 50 | 51 | solve(a==L, grad_sol, bcs=bc) 52 | sol.vector().axpy(-alpha, grad_sol) 53 | 54 | err_vec = assemble( dot(grad(sol), grad(v)) * dx - f * v * dx, bcs=bc) 55 | with err_vec.dat.vec_ro as vv: 56 | err = vv.norm() 57 | it += 1 58 | if verbose: 59 | print(f"It {it:4}, error={err:2.4e}") 60 | if err > 1e20: return 0.0 61 | return it 62 | 63 | its_l2 = [] 64 | its_L2 = [] 65 | its_H1 = [] 66 | its_H01 = [] 67 | Ns = [2,5,10,20,40,80] 68 | for N in Ns: 69 | print("==========================") 70 | print("Solving for N =", N) 71 | print("==========================") 72 | it_l2 = solvePoisson(N, "l2", alpha=alpha, verbose=verbose) 73 | it_L2 = solvePoisson(N, "L2", alpha=alpha, verbose=verbose) 74 | it_H1 = solvePoisson(N, "H1", alpha=alpha, verbose=verbose) 75 | it_H01 = solvePoisson(N, "H01", alpha=alpha, verbose=verbose) 76 | its_l2.append(it_l2) 77 | its_L2.append(it_L2) 78 | its_H1.append(it_H1) 79 | its_H01.append(it_H01) 80 | 81 | print("l2:", its_l2) 82 | print("L2:", its_L2) 83 | print("H1:", its_H1) 84 | print("H01:", its_H01) 85 | plt.semilogx(Ns, its_l2, label="l2") 86 | plt.semilogx(Ns, its_L2, label="L2") 87 | plt.semilogx(Ns, its_H1, label="H1") 88 | plt.semilogx(Ns, its_H01, label="H01") 89 | plt.legend() 90 | plt.show() 91 | -------------------------------------------------------------------------------- /chapters/introduction.tex: -------------------------------------------------------------------------------- 1 | \section*{Introduction} 2 | 3 | These notes started as backup material for a course on some deeper topics in mathematical engineering at Pontificia Universidad Católica de Chile, the second semester of 2024, but then evolved to include more introductory material on Numerical Analysis of PDEs and Continuum Mechanics for our course \emph{Applications of Functional Analysis and PDEs in Engineering}. The hand-written notes by Federico Fuentes were absolutely fundamental to provide depth and context to most of the content presented. The final cornerstone of these notes was the typesetting effort done by Bastián Herrera, whose knowledge on these topics has provided a constant input for perfectioning the contents and the coherence of the presentation. We also thank the School of Engineering at Pontificia Universidad Católica de Chile for the funding. In general, these notes put together contents found online and in books, instead of providing original material and new proofs of theorems. Beyond the books being cited, which are in general fantastic references, we tried to acknowledge many of the available notes online that provide fantastic level of detail and insight. 4 | 5 | The idea of these notes is to provide mathematical tools for students that give them the ability to assess the difficulty of mathematical problems, mainly within the world of partial differential equations (PDEs). The target is ultimately to implement these models, so that all tools are oriented towards having solid foundations that allow one to trust a computational model. Informally speaking, the main mathematical concepts to haunt us throughout all these notes are: 6 | \begin{itemize} 7 | \item Existence and uniqueness: it is a natural baseline in the mathematician's world to try to solve only problems that \emph{have} a solution. Otherwise, things might be as pointless as developing an iterative method for finding real numbers such that $x^2 = -1$. Uniqueness is a further luxury, but sometimes two different methods give two different solutions, and having only those things at hand can make it difficult to distinguish whether that is a bug or a feature of the model. There exist some root-isolation methods that allow to find solutions of a problem that are \emph{different} from a given one. This is out of the scope of this course. 8 | \item Stability: the intuitive idea behind this is that small perturbations in the data give rise to small changes in the solution. This typically looks like 9 | \begin{equation*} 10 | \| u\|_X \leq C\| f\|_{X'}, 11 | \end{equation*} 12 | where $u$ is the solution of a problem that depends on the data $f$, and $X$ is some functional (hopefully Hilbert) space with dual $X'$. More rigorously, this means that the solution map $f \mapsto u(f)$ is bounded, or continuous in the linear case. Stability also sometimes refers to time dynamics and the fact that a discrete solution stays \emph{within a certain distance} of the true solution throughout a simulation. In the continuous setting, it might also mean that there are no finite-time singularities. In general, stability is not a well-defined term, but still a widely understood one to anyone who has struggled to get a code to run correctly, and a highly desired property. 13 | \end{itemize} 14 | All other properties (or at least most of them anyway) are ways to guarantee that a problem enjoys one of these nice properties. There are ways to handle problems that do not have those properties, but they are almost always extremely problem-dependent, and the person studying such problems should dive deep into the sectorial knowledge to see how certain communities deal with such issues. This is an aspect that mathematically-oriented people almost always disregard, which has some severe mathematical (and social) consequences. In fact, some extremely classical models in engineering are still far from understood mathematically, such as the Navier-Stokes equations. This has not prevented the computational fluid dynamics (CFD) community from solving these models with extreme efficiency, and from further leveraging them for industrial applications which, unsurprisingly, work fantastically. Discovering the amazing ways in which mathematically agnostic communities solve mathematically hard problems is, and will probably be for very long, a beautiful opportunity for collaboration. 15 | 16 | 17 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | ## Core latex/pdflatex auxiliary files: 2 | *.aux 3 | *.lof 4 | *.log 5 | *.lot 6 | *.fls 7 | *.out 8 | *.toc 9 | *.fmt 10 | *.fot 11 | *.cb 12 | *.cb2 13 | .*.lb 14 | 15 | ## Intermediate documents: 16 | *.dvi 17 | *.xdv 18 | *-converted-to.* 19 | 20 | # these rules might exclude image files for figures etc. 21 | *.ps 22 | *.eps 23 | *.pdf 24 | 25 | ## Bibliography auxiliary files (bibtex/biblatex/biber): 26 | *.bbl 27 | *.bcf 28 | *.blg 29 | *-blx.aux 30 | *-blx.bib 31 | *.run.xml 32 | 33 | ## Build tool auxiliary files: 34 | *.fdb_latexmk 35 | *.synctex 36 | *.synctex(busy) 37 | *.synctex.gz 38 | *.synctex.gz(busy) 39 | *.pdfsync 40 | 41 | ## Build tool directories for auxiliary files 42 | # latexrun 43 | latex.out/ 44 | 45 | ## Auxiliary and intermediate files from other packages: 46 | # algorithms 47 | *.alg 48 | *.loa 49 | 50 | # achemso 51 | acs-*.bib 52 | 53 | # amsthm 54 | *.thm 55 | 56 | # beamer 57 | *.nav 58 | *.pre 59 | *.snm 60 | *.vrb 61 | 62 | # changes 63 | *.soc 64 | 65 | # comment 66 | *.cut 67 | 68 | # cprotect 69 | *.cpt 70 | 71 | # elsarticle (documentclass of Elsevier journals) 72 | *.spl 73 | 74 | # endnotes 75 | *.ent 76 | 77 | # fixme 78 | *.lox 79 | 80 | # feynmf/feynmp 81 | *.mf 82 | *.mp 83 | *.t[1-9] 84 | *.t[1-9][0-9] 85 | *.tfm 86 | 87 | #(r)(e)ledmac/(r)(e)ledpar 88 | *.end 89 | *.?end 90 | *.[1-9] 91 | *.[1-9][0-9] 92 | *.[1-9][0-9][0-9] 93 | *.[1-9]R 94 | *.[1-9][0-9]R 95 | *.[1-9][0-9][0-9]R 96 | *.eledsec[1-9] 97 | *.eledsec[1-9]R 98 | *.eledsec[1-9][0-9] 99 | *.eledsec[1-9][0-9]R 100 | *.eledsec[1-9][0-9][0-9] 101 | *.eledsec[1-9][0-9][0-9]R 102 | 103 | # glossaries 104 | *.acn 105 | *.acr 106 | *.glg 107 | *.glo 108 | *.gls 109 | *.glsdefs 110 | *.lzo 111 | *.lzs 112 | *.slg 113 | *.slo 114 | *.sls 115 | 116 | # uncomment this for glossaries-extra (will ignore makeindex's style files!) 117 | # *.ist 118 | 119 | # gnuplot 120 | *.gnuplot 121 | *.table 122 | 123 | # gnuplottex 124 | *-gnuplottex-* 125 | 126 | # gregoriotex 127 | *.gaux 128 | *.glog 129 | *.gtex 130 | 131 | # htlatex 132 | *.4ct 133 | *.4tc 134 | *.idv 135 | *.lg 136 | *.trc 137 | *.xref 138 | 139 | # hyperref 140 | *.brf 141 | 142 | # knitr 143 | *-concordance.tex 144 | # TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files 145 | # *.tikz 146 | *-tikzDictionary 147 | 148 | # listings 149 | *.lol 150 | 151 | # luatexja-ruby 152 | *.ltjruby 153 | 154 | # makeidx 155 | *.idx 156 | *.ilg 157 | *.ind 158 | 159 | # minitoc 160 | *.maf 161 | *.mlf 162 | *.mlt 163 | *.mtc[0-9]* 164 | *.slf[0-9]* 165 | *.slt[0-9]* 166 | *.stc[0-9]* 167 | 168 | # minted 169 | _minted* 170 | *.pyg 171 | 172 | # morewrites 173 | *.mw 174 | 175 | # newpax 176 | *.newpax 177 | 178 | # nomencl 179 | *.nlg 180 | *.nlo 181 | *.nls 182 | 183 | # pax 184 | *.pax 185 | 186 | # pdfpcnotes 187 | *.pdfpc 188 | 189 | # sagetex 190 | *.sagetex.sage 191 | *.sagetex.py 192 | *.sagetex.scmd 193 | 194 | # scrwfile 195 | *.wrt 196 | 197 | # svg 198 | svg-inkscape/ 199 | 200 | # sympy 201 | *.sout 202 | *.sympy 203 | sympy-plots-for-*.tex/ 204 | 205 | # pdfcomment 206 | *.upa 207 | *.upb 208 | 209 | # pythontex 210 | *.pytxcode 211 | pythontex-files-*/ 212 | 213 | # tcolorbox 214 | *.listing 215 | 216 | # thmtools 217 | *.loe 218 | 219 | # TikZ & PGF 220 | *.dpth 221 | *.md5 222 | *.auxlock 223 | 224 | # titletoc 225 | *.ptc 226 | 227 | # todonotes 228 | *.tdo 229 | 230 | # vhistory 231 | *.hst 232 | *.ver 233 | 234 | # easy-todo 235 | *.lod 236 | 237 | # xcolor 238 | *.xcp 239 | 240 | # xmpincl 241 | *.xmpi 242 | 243 | # xindy 244 | *.xdy 245 | 246 | # xypic precompiled matrices and outlines 247 | *.xyc 248 | *.xyd 249 | 250 | # endfloat 251 | *.ttt 252 | *.fff 253 | 254 | # Latexian 255 | TSWLatexianTemp* 256 | 257 | ## Editors: 258 | # WinEdt 259 | *.bak 260 | *.sav 261 | 262 | # Texpad 263 | .texpadtmp 264 | 265 | # LyX 266 | *.lyx~ 267 | 268 | # Kile 269 | *.backup 270 | 271 | # gummi 272 | .*.swp 273 | 274 | # KBibTeX 275 | *~[0-9]* 276 | 277 | # TeXnicCenter 278 | *.tps 279 | 280 | # auto folder when using emacs and auctex 281 | ./auto/* 282 | *.el 283 | 284 | # expex forward references with \gathertags 285 | *-tags.tex 286 | 287 | # standalone packages 288 | *.sta 289 | 290 | # Makeindex log files 291 | *.lpz 292 | 293 | # xwatermark package 294 | *.xwm 295 | 296 | # REVTeX puts footnotes in the bibliography by default, unless the nofootinbib 297 | # option is specified. Footnotes are the stored in a file with suffix Notes.bib. 298 | # Uncomment the next line to have this generated file ignored. 299 | #*Notes.bib 300 | 301 | 302 | output/ 303 | -------------------------------------------------------------------------------- /src/macros.tex: -------------------------------------------------------------------------------- 1 | \usepackage[utf8]{inputenc} 2 | \usepackage{amsmath, amsthm, amssymb, mathpazo, isomath, mathtools} 3 | \usepackage{thmtools} 4 | \usepackage{subcaption,graphicx} 5 | \usepackage{fullpage} 6 | \usepackage{booktabs} 7 | \usepackage{hyperref} 8 | \usepackage{environ} 9 | \usepackage[most]{tcolorbox} 10 | %%%% Remove some annoying hyperref warnings. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 11 | % See https://tex.stackexchange.com/questions/10555/hyperref-warning-token-not-allowed-in-a-pdf-string 12 | \makeatletter 13 | \pdfstringdefDisableCommands{\let\HyPsd@CatcodeWarning\@gobble} 14 | \makeatother 15 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 16 | 17 | \usepackage{algorithm} 18 | \usepackage{algpseudocode} 19 | \usepackage{mathtools} 20 | \usepackage{todonotes} 21 | \usepackage{cancel} 22 | \usepackage{pgfplots} 23 | \pgfplotsset{compat=1.18} 24 | 25 | \DeclareMathOperator{\spanned}{\text{span}} 26 | \DeclareMathOperator{\grad}{\nabla} 27 | \DeclareMathOperator{\Grad}{\text{Grad}} 28 | \DeclareMathOperator{\Cof}{\text{Cof}} 29 | \DeclareMathOperator{\dive}{\text{div}} 30 | \DeclareMathOperator{\Dive}{\text{Div}} 31 | \DeclareMathOperator{\curl}{\text{curl}} 32 | \DeclareMathOperator{\Curl}{\text{Curl}} 33 | \DeclareMathOperator{\tr}{\text{tr}} 34 | \DeclareMathOperator{\inv}{\text{Inv}} 35 | \DeclareMathOperator{\dom}{\text{dom}} 36 | \DeclareMathOperator{\supp}{\text{supp}} 37 | \DeclareMathOperator{\diag}{\text{diag}} 38 | \DeclareMathOperator{\bolddiag}{\textbf{diag}} 39 | \DeclareMathOperator{\im}{\text{im}} 40 | \DeclareMathOperator{\dist}{\text{dist}} 41 | \DeclareMathOperator{\rank}{\text{rank}} 42 | \DeclareMathOperator{\interior}{\text{int}} 43 | 44 | 45 | \declaretheoremstyle[ 46 | spaceabove=6pt, spacebelow=12pt, 47 | headfont=\normalfont\bfseries, 48 | notefont=\mdseries, notebraces={(}{)}, 49 | bodyfont=\normalfont, 50 | postheadspace=1em, 51 | ]{thmstyle} 52 | \declaretheoremstyle[ 53 | spaceabove=6pt, spacebelow=12pt, 54 | headfont=\normalfont\bfseries, 55 | notefont=\mdseries, notebraces={(}{)}, 56 | bodyfont=\normalfont, 57 | postheadspace=1em, 58 | ]{lemmastyle} 59 | \declaretheoremstyle[ 60 | spaceabove=6pt, spacebelow=12pt, 61 | headfont=\normalfont\bfseries, 62 | notefont=\mdseries, notebraces={(}{)}, 63 | bodyfont=\normalfont, 64 | postheadspace=0.5em, 65 | ]{defstyle} 66 | \declaretheoremstyle[ 67 | spaceabove=6pt, spacebelow=12pt, 68 | headfont=\normalfont\bfseries, 69 | notefont=\mdseries, notebraces={(}{)}, 70 | bodyfont=\normalfont, 71 | postheadspace=1em, 72 | ]{smallstyle} 73 | 74 | \declaretheorem{theorem}[style=thmstyle, parent=chapter, name=Theorem] 75 | \declaretheorem{lemma}[style=thmstyle, sibling=theorem, name=Lemma] 76 | \declaretheorem{remark}[style=smallstyle, sibling=theorem, name=Remark] 77 | \declaretheorem{definition}[style=defstyle, sibling=theorem, name=Definition] 78 | \declaretheorem{corollary}[style=smallstyle, sibling=theorem, name=Corollary] 79 | \declaretheorem{note}[style=smallstyle, sibling=theorem, name=Note] 80 | 81 | \newcommand{\R}{\mathbb{R}} 82 | \newcommand{\C}{\mathbb{C}} 83 | \newcommand{\D}{\mathcal{D}} 84 | \newcommand{\T}{\mathcal{T}} 85 | \newcommand{\N}{\mathbb{N}} 86 | \newcommand{\tenF}{\ten{F}} 87 | \newcommand{\vX}{\nabla_X} 88 | \newcommand{\vx}{\nabla_x} 89 | \newcommand{\vvarphi}{\vec{\varphi}} 90 | \newcommand{\RightComment}[1]{\hfill \(\triangleright\) \textit{#1}} 91 | \newcommand{\example}[1]{ \begin{tcolorbox}[colback=blue!5!white,colframe=blue!75!black!70!white,title=Example]#1 \end{tcolorbox}} 92 | \newcommand{\nota}[1]{{\colorbox{red!30}{[#1]}}} 93 | 94 | \newcommand{\mat}{\matrixsym} 95 | \newcommand{\ten}{\tensorsym} 96 | \newcommand{\parder}[2]{\frac{\partial #1}{\partial #2} } 97 | \newcommand{\tin}{\text{ in }} 98 | \newcommand{\ton}{\text{ on }} 99 | 100 | \renewcommand{\P}{\mathcal{P}} 101 | \renewcommand{\vec}{\vectorsym} 102 | 103 | \usepackage{listings} 104 | \usepackage{xcolor} 105 | \definecolor{codegreen}{rgb}{0,0.6,0} 106 | \definecolor{codegray}{rgb}{0.5,0.5,0.5} 107 | \definecolor{codepurple}{rgb}{0.58,0,0.82} 108 | \definecolor{backcolour}{rgb}{0.95,0.95,0.92} 109 | \lstdefinestyle{mystyle}{ 110 | backgroundcolor=\color{backcolour}, commentstyle=\color{codegreen}, 111 | keywordstyle=\color{magenta}, 112 | numberstyle=\tiny\color{codegray}, 113 | stringstyle=\color{codepurple}, 114 | basicstyle=\ttfamily\footnotesize, 115 | breakatwhitespace=false, 116 | % breaklines=false, 117 | captionpos=b, 118 | keepspaces=true, 119 | numbers=none, 120 | numbersep=6pt, 121 | showspaces=false, 122 | showstringspaces=false, 123 | showtabs=false, 124 | tabsize=2, 125 | % frameround=tttn, 126 | framerule=1.6pt, 127 | rulecolor=\color{red!60!black} 128 | } 129 | \lstset{style=mystyle} 130 | 131 | \def\tightTOP{6pt} 132 | \def\tightBOTTOM{6pt} 133 | \def\tightLEFT{6pt} 134 | \def\tightRIGHT{6pt} 135 | \NewEnviron{tightalign*}{% 136 | \setlength{\abovedisplayskip}{\tightTOP}% 137 | \setlength{\belowdisplayskip}{\tightBOTTOM}% 138 | \setlength{\abovedisplayshortskip}{\tightLEFT}% 139 | \setlength{\belowdisplayshortskip}{\tightRIGHT}% 140 | \begin{align*} 141 | \BODY % \BODY is the content of the environment 142 | \end{align*} 143 | } 144 | 145 | \NewEnviron{tightalign}{% 146 | \setlength{\abovedisplayskip}{\tightTOP}% 147 | \setlength{\belowdisplayskip}{\tightBOTTOM}% 148 | \setlength{\abovedisplayshortskip}{\tightLEFT}% 149 | \setlength{\belowdisplayshortskip}{\tightRIGHT}% 150 | \begin{align} 151 | \BODY % \BODY is the content of the environment 152 | \end{align} 153 | } 154 | -------------------------------------------------------------------------------- /tareas/tarea1/main.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \usepackage[utf8]{inputenc} 3 | \usepackage{amsmath, amsthm, amssymb, mathpazo, isomath, mathtools} 4 | \usepackage{subcaption,graphicx,pgfplots} 5 | \usepackage{fullpage} 6 | \usepackage{booktabs} 7 | \usepackage{hyperref} 8 | \usepackage{algorithm, algorithmic} 9 | \usepackage{mathtools} 10 | \usepackage{todonotes} 11 | 12 | \title{Tarea 1} 13 | %\author{Nicol\'as A Barnafi\thanks{Instituto de Ingeniería Biológica y Médica, Pontificia Universidad Católica de Chile, Chile}, Axel Osses\thanks{Departamento de Ingeniería Matemática, Universidad de Chile, Chile}} 14 | %\author{Nicol\'as A Barnafi} 15 | \date{} 16 | 17 | \renewcommand{\vec}{\vectorsym} 18 | \newcommand{\mat}{\matrixsym} 19 | \newcommand{\ten}{\tensorsym} 20 | \DeclareMathOperator{\grad}{\nabla} 21 | \DeclareMathOperator{\dive}{\text{div}} 22 | \DeclareMathOperator{\curl}{\text{curl}} 23 | \DeclareMathOperator{\tr}{\text{tr}} 24 | \DeclareMathOperator{\sym}{\text{sym}} 25 | \newtheorem{remark}{Remark} 26 | \newtheorem{definition}{Definition} 27 | \newcommand{\R}{\mathbb{R}} 28 | \newcommand{\D}{\mathcal{D}} 29 | 30 | \newcommand{\tin}{\text{in}} 31 | \newcommand{\ton}{\text{on}} 32 | 33 | \newtheorem{theorem}{Theorem} 34 | \newtheorem{lemma}{Lemma} 35 | \newcommand{\pts}[1]{[{\bf #1 puntos}]} 36 | 37 | \begin{document} 38 | 39 | \maketitle 40 | \hfill \textbf{Fecha de entrega: 23:59 del 11/04/2025} 41 | 42 | \todo[inline,color=white!90!black]{\textbf{Instrucciones: } La tarea debe ser entregada de manera individual en un informe en formato .pdf a través del buzón habilitado en la plataforma Canvas, donde deben mostrar también el código desarrollado. Para su conveniencia, pueden entregar las tareas en un Jupyter Notebook, de modo que sea más cómodo mostrar el código. La política de atrasos será: se calculará un factor lineal que vale 1 a la hora de entrega y 0 48 horas después. Esto multiplicará su puntaje obtenido. Pueden usar ChatGPT u otros modelos solo a conciencia. El uso de salidas de GPT sin su debida comprensión será severamente sancionado. } 43 | 44 | \begin{enumerate} 45 | \item Considere el dominio $\Omega=(a,b)$ con $a0$. 52 | \end{itemize} 53 | Para este modelo: 54 | \begin{enumerate} 55 | \item\pts{1} Proponga un esquema de discretización y demuestre que es consistente. 56 | \item\pts{1} Usando el análisis de von Neumann, proponga hipótesis de la solución discreta y los parámetros de discretización que permitan obtener una aproximación estable, y por lo tanto convergente. 57 | \item\pts{1} Proponga tres formulaciones de discretización en tiempo: (i) explícita, (ii) implícita, y (iii) semi-implícita. 58 | \item\pts{1} Formule un método iterativo (Newton o punto fijo) para resolver la formulación implícita y comente sobre las propiedades de convergencia que tiene el método propuesto. 59 | \item\pts{3} Implemente el método implícito propuesto y grafique la evolución de la solución para $\mu=0.1$. Comente sus resultados. 60 | \item\pts{2} Implemente el método semi-implícito propuesto y grafique la evolución de la solución para $\mu=0.1$. Comente sus resultados y compárelos con los obtenidos en el punto anterior. 61 | \end{enumerate} 62 | 63 | \item En esta pregunta, jugaremos con la definición de las distribuciones. Usaremos la notación $\mathcal D(\R) = C_0^\infty(\R)$, espacio de funciones suaves con soporte compacto en $\R$. 64 | \begin{enumerate} 65 | \item\pts{1} Definimos la inyección de funciones integrables a distribuciones $T:L^1(\R) \to \left(\mathcal D(\R)\right)'$ según la acción 66 | $$ (Tf)(\varphi) = \left\langle f, \varphi\right\rangle_{\mathcal D'\times \mathcal D}.$$ 67 | Demuestre que dicha inyección es continua. 68 | \item\pts{1}Caracterice la convergencia en el espacio de distribuciones $\left(\mathcal D(\R)\right)'$. 69 | \item\pts{1}Demuestre que, dado $x_0\in \R$, la sucesión inducida por la función $I_n(x) = nI_{(x_0-1/2n, x_0+1/2n)}$, i.e. $\{I_n\}_n$, converge como distribución a la delta de Dirac $\delta_{x_0}$. 70 | \item\pts{1}Demuestre que no existe ninguna función $f$ en $L^1$ tal que $T_f = \delta_0$. 71 | \item\pts{2} Sabemos que dada una función vectorial $F:\Omega \to \R^d$, su divergencia está dada por $\dive F=\sum_i \partial_{x_i}F$. Para extender esta noción a distribuciones, demuestre dado un dominio $\Omega$ los espacios $(\mathcal D(\Omega, \R^d))'$ y $[(\mathcal D(\Omega))']^d$ son homeomorfos\footnote{$X^3 = X\times X\times X$} y con ello construya una definición de divergencia en $(\mathcal D(\Omega, \R^d))'$. Hint: $\int_\Omega (\dive F)\varphi = \int_{\partial\Omega} \varphi F\cdot \vec n - \int_\Omega F\cdot \grad \varphi$. 72 | \item\pts{2} Extienda la construcción de la pregunta anterior para definir un $\curl$ distribucional. 73 | \item\pts{2} Sea $\vec e_j$ el $j$-ésimo vector canónico en $\R^d$. Definimos el operador de diferencias finitas parciales de paso $h$ como 74 | $$ D_j^hf(\vec x) \coloneqq \frac{f(x+h\vec e_j) - f(x)}{h}. $$ 75 | Demuestre que para cada $f$ en $\mathcal D(\R^d)$, se tiene que $D_j^hf $ converge a $\frac{\partial f}{\partial x_j}$ en la topología de $\mathcal D(\R^d)$ cuando $h$ va a 0. 76 | \item\pts{2} Considere la función $\Phi:\R^3\to \mathbb C$ dada por 77 | $$ \Phi(x) = \frac{1}{4\pi|x|} e^{-ik |x|}. $$ 78 | Muestre que se tiene la siguiente igualdad en $\mathcal D'(\R^3)$: 79 | $$ -\Delta \Phi - k^2 \Phi = \delta_0, $$ 80 | i.e. en el sentido de las distribuciones, donde $\delta_0$ es la delta de Dirac en $x=0$. 81 | \end{enumerate} 82 | \end{enumerate} 83 | 84 | \todo[inline,color=white!90!black]{\textbf{Nota: } Abriremos un foro en Canvas para revisar cualquier typo y/o error que haya en el enunciado.} 85 | \end{document} 86 | 87 | -------------------------------------------------------------------------------- /tareas/tarea5/main.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \usepackage[utf8]{inputenc} 3 | \usepackage{amsmath, amsthm, amssymb, mathpazo, isomath, mathtools} 4 | \usepackage{subcaption,graphicx,pgfplots} 5 | \usepackage{fullpage} 6 | \usepackage{booktabs} 7 | \usepackage{hyperref} 8 | \usepackage{algorithm, algorithmic} 9 | \usepackage{mathtools} 10 | \usepackage{todonotes} 11 | 12 | \title{Tarea 5} 13 | %\author{Nicol\'as A Barnafi\thanks{Instituto de Ingeniería Biológica y Médica, Pontificia Universidad Católica de Chile, Chile}, Axel Osses\thanks{Departamento de Ingeniería Matemática, Universidad de Chile, Chile}} 14 | %\author{Nicol\'as A Barnafi} 15 | \date{} 16 | 17 | \renewcommand{\vec}{\vectorsym} 18 | \newcommand{\mat}{\matrixsym} 19 | \newcommand{\ten}{\tensorsym} 20 | \DeclareMathOperator{\grad}{\nabla} 21 | \DeclareMathOperator{\dive}{\text{div}} 22 | \DeclareMathOperator{\curl}{\text{curl}} 23 | \DeclareMathOperator{\tr}{\text{tr}} 24 | \DeclareMathOperator{\sym}{\text{sym}} 25 | \newtheorem{remark}{Remark} 26 | \newtheorem{definition}{Definition} 27 | \newcommand{\R}{\mathbb{R}} 28 | \newcommand{\D}{\mathcal{D}} 29 | \newcommand{\parder}[2]{\frac{\partial\,#1}{\partial\,#2}} 30 | 31 | \newcommand{\tin}{\text{in}} 32 | \newcommand{\ton}{\text{on}} 33 | 34 | \newtheorem{theorem}{Theorem} 35 | \newtheorem{lemma}{Lemma} 36 | \newcommand{\pts}[1]{[{\bf #1 puntos}] } 37 | 38 | \begin{document} 39 | 40 | \maketitle 41 | \hfill \textbf{Fecha de entrega: 23:59 del 13/06/2025} 42 | 43 | \todo[inline,color=white!90!black]{\textbf{Instrucciones: } La tarea debe ser entregada de manera individual en un informe en formato .pdf a través del buzón habilitado en la plataforma Canvas, donde deben mostrar también el código desarrollado. Para su conveniencia, pueden entregar las tareas en un Jupyter Notebook, de modo que sea más cómodo mostrar el código. En cualquier caso, se debe entregar un único archivo como respuesta a la tarea. La política de atrasos será: se calculará un factor lineal que vale 1 a la hora de entrega y 0 48 horas después. Esto multiplicará su puntaje obtenido. Pueden usar ChatGPT u otros modelos solo a conciencia. El uso de salidas de GPT sin su debida comprensión será severamente sancionado. } 44 | 45 | \begin{enumerate} 46 | \item Considere la deformación dada por 47 | \begin{align*} 48 | x_1(\vec X, t) &= e^tX_1 - e^{-t} X_2 \\ 49 | x_2(\vec X, t) &= e^t X_1 + e^{-t}X_2 \\ 50 | x_3(\vec X, t) &= X_3. 51 | \end{align*} 52 | \begin{itemize} 53 | \item\pts{1} Calcule el campo de desplazamiento, la velocidad, y la aceleración en coordenadas materiales (de referencia). 54 | \item\pts{1} Calcule el campo de desplazamiento, la velocidad y la aceleración en coordenadas espaciales (deformadas). 55 | \item\pts{1} Calcule los tensores $\ten F$, $\ten C$, y $\ten E$. 56 | \item\pts{1} Ignore la tercera componente y considere $\Omega$ un cuadrado unitario. Grafique en Python la configuración inicial ($t=0$), y la configuración deformada en los instantes 0.5, 1.0 y 2.0. 57 | \item\pts{1} En los instantes descritos, grafique la norma de Frobenius de $\ten C$ en configuración de referencia y en la deformada. Comente cómo cambia la interpretación del tensor entre ambas configuraciones, y proponga una manera de modificar el tensor $\ten C$ para que tenga una mejor interpretación en la configuración deformada. 58 | \end{itemize} 59 | \begin{itemize} 60 | \item\pts{1} Dada una matriz $\mat A$ se definen 61 | $$I_1(\mat A) = \tr \mat A,$$ 62 | $$I_2(\mat A) = \tr \left(\det (\mat A) \mat A^{-T}\right) = \frac 1 2 \left([\tr \mat A]^2 - \tr(\mat A^2)\right),$$ 63 | $$I_3(\mat A) = \det A.$$ 64 | 65 | Muestre que si se hace un cambio de base a la matriz $\mat A$ en el sentido de reemplazarla por $\mat Q^{-1} \mat A \mat Q$, estas funciones no cambian su valor, i.e. $I_i(\mat A) = I_i(\mat Q^{-1} \mat A\mat Q)$. Por esta razón se les llama \emph{invariantes} de $\mat A$. Hint: Para el determinante podría ser útil investigar sobre el símbolo de Levi-Civita $\epsilon_{ijk}$ y su uso en el cálculo del determinante. 66 | \item\pts{1} El Teorema de Caley-Hamilton establece que toda matriz satisface su propia ecuación característica usando las invariantes: 67 | $$ \mat A^3 - I_1(\mat A) \mat A^2 + I_2(\mat A) \mat A - I_3(\mat A) \mat I = 0. $$ 68 | Derive esta ecuación con respecto a $\mat A$ y desarrolle la ecuación para demostrar que $\parder{\det \mat A}{\mat A} = \det (\mat A) \mat A^{-T}$. 69 | \end{itemize} 70 | \item Considere un material hiperelástico Neo-Hookeano, donde la energía hiperelástica está dada por 71 | $$ \Psi(\ten F) = \frac C 2(\tr \ten C - 3) + \frac \kappa 2 \left( J -1\right)^2. $$ 72 | \begin{itemize} 73 | \item\pts{1} Calcule el tensor de Piola. 74 | \item\pts{1} Escriba la formulación débil asociada a la ecuación de momentum con el tensor explícito de Piola calculado. 75 | \end{itemize} 76 | \item Modifique la deducción de la conservación de masa para considerar un flujo de masa dado por su gradiente normal, i.e. considere un término adicional de superficie para todo subdominio $\omega_t\subseteq\Omega_t$: 77 | $$ \int_{\partial \omega_t} \ten K \grad\rho \cdot \vec n \,dS, $$ 78 | \begin{itemize} 79 | \item\pts{1} Muestre la ecuación resultante, y muestre que el operador diferencial en espacio es un operador ADR. 80 | \end{itemize} 81 | \item Considerar ecuación de momentum lineal de referencia: 82 | $$ \rho_0\ddot{\vec u} - \dive_X \ten P(\ten F)=0. $$ 83 | \begin{itemize} 84 | \item\pts{1} Cambie de variables el tensor $\ten P$ para que dependa de $\ten E$, y luego linealice con respecto a $\vec u = 0$. Recuerde que linealizar una ecuación 85 | $$ F(x) = 0 $$ 86 | con respecto a un punto $x_0$ significa reemplazar $F(x) \approx F(x_0) + dF(x_0)[\delta x]$ para obtener una ecuación para el incremento $\delta x$. Acá, $dF(x_0)[\delta x]$ es la derivada de Gateaux en dirección $\delta x$. 87 | \item\pts{1} Muestre que el problema resultante es el problema de elastodinámica lineal, e identifique claramente el tensor de Hooke que obtiene. 88 | \end{itemize} 89 | 90 | \item\pts{1} Defina su proyecto de curso. 91 | \end{enumerate} 92 | 93 | \todo[inline,color=white!90!black]{\textbf{Nota: } Abriremos un foro en Canvas para revisar cualquier typo y/o error que haya en el enunciado.} 94 | \end{document} 95 | 96 | -------------------------------------------------------------------------------- /tareas/midterm/main.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \usepackage[utf8]{inputenc} 3 | \usepackage{amsmath, amsthm, amssymb, mathpazo, isomath, mathtools} 4 | \usepackage{subcaption,graphicx,pgfplots} 5 | \usepackage{fullpage} 6 | \usepackage{booktabs} 7 | \usepackage{hyperref} 8 | \usepackage{algorithm, algorithmic} 9 | \usepackage{mathtools} 10 | \usepackage{todonotes} 11 | 12 | \title{Examen} 13 | %\author{Nicol\'as A Barnafi\thanks{Instituto de Ingeniería Biológica y Médica, Pontificia Universidad Católica de Chile, Chile}, Axel Osses\thanks{Departamento de Ingeniería Matemática, Universidad de Chile, Chile}} 14 | %\author{Nicol\'as A Barnafi} 15 | \date{} 16 | 17 | \renewcommand{\vec}{\vectorsym} 18 | \newcommand{\mat}{\matrixsym} 19 | \newcommand{\ten}{\tensorsym} 20 | \DeclareMathOperator{\grad}{\nabla} 21 | \DeclareMathOperator{\dive}{\text{div}} 22 | \DeclareMathOperator{\curl}{\text{curl}} 23 | \DeclareMathOperator{\tr}{\text{tr}} 24 | \DeclareMathOperator{\sym}{\text{sym}} 25 | \newtheorem{remark}{Remark} 26 | \newtheorem{definition}{Definition} 27 | \newcommand{\R}{\mathbb{R}} 28 | \newcommand{\D}{\mathcal{D}} 29 | \newcommand{\parder}[2]{\frac{\partial\,#1}{\partial\,#2}} 30 | 31 | \newcommand{\tin}{\text{in}} 32 | \newcommand{\ton}{\text{on}} 33 | 34 | \newtheorem{theorem}{Theorem} 35 | \newtheorem{lemma}{Lemma} 36 | \newcommand{\pts}[1]{[{\bf #1 puntos}] } 37 | 38 | \begin{document} 39 | 40 | \maketitle 41 | \hfill \textbf{Fecha de entrega: 23:59 del 30/06/2025} 42 | 43 | \todo[inline,color=white!90!black]{\textbf{Instrucciones: } La tarea debe ser entregada de manera individual en un informe en formato .pdf a través del buzón habilitado en la plataforma Canvas, donde deben mostrar también el código desarrollado. Para su conveniencia, pueden entregar las tareas en un Jupyter Notebook, de modo que sea más cómodo mostrar el código. En cualquier caso, se debe entregar un único archivo como respuesta a la tarea. Entregas atrasadas tienen un 1.0 automáticamente. Pueden usar ChatGPT u otros modelos solo a conciencia. El uso de salidas de GPT sin su debida comprensión será severamente sancionado. Todas las preguntas tienen un punto, y el puntaje estara dado por: 1.0 punto si la pregunta esta correcta, 0.5 si tiene fallas menores pero esta principalmente bien, 0.0 si presenta errores conceptuales importantes.} 44 | 45 | \begin{enumerate} 46 | 47 | \item Considere el problema ADR con condiciones de borde homogéneas: 48 | $$ \begin{aligned} 49 | -\mu \Delta u + \vec b \cdot \grad u + c u &= f \qquad \Omega \\ 50 | u &= 0 \qquad \partial\Omega. 51 | \end{aligned}$$ 52 | Asumiendo que $\mu,\vec b,c$ son funciones, considere la discretización de este problema en el dominio $\Omega = (0,1)$. 53 | \begin{itemize} 54 | \item\pts{1} Muestre que, bajo hipótesis adecuadas sobre los parámetros, este problema tiene una única solución débil y que además esta es continua. 55 | \item\pts{1}Describa cómo aproximar este problema con diferencias finitas y escriba explícitamente la matriz discreta. 56 | \item\pts{1}Describa cómo aproximar este problema con elementos finitos (de primer orden) y escriba explícitamente la matriz discreta. 57 | \item\pts{1}Existe algún rango de parámetros donde las discretizaciones resulten en el mismo problema? Y parámetros donde sean diferentes? 58 | \item\pts{1}Explique las garantías teóricas que tienen diferencias finitas y elementos finitos, y compárelas. Qué le parece más conveniente en este caso? 59 | \end{itemize} 60 | 61 | \item Considere el problema biarmónico dado por el bilaplaciano: 62 | $$ \Delta^2 u = f \qquad \Omega, $$ 63 | con condición de borde $u = \grad u\cdot \vec n = 0$ en $\partial\Omega$, y considere el espacio funcional 64 | $$ H_0^2(\Omega) = \{v \in H^2(\Omega): v=\grad v \cdot \vec n = 0 \quad \partial\Omega\}. $$ 65 | \begin{itemize} 66 | \item\pts{1} Demuestre que $v\mapsto|\Delta v|$ es una norma . 67 | \item\pts{1} Demuestre que $v\mapsto|\Delta v|$ es una norma equivalente a la norma natural de $H^2(\Omega)$ en $H_0^2(\Omega)$. 68 | \item\pts{1} Encuentre la formulación variacional del problema y diga cuál es la regularidad mínima que requiere para $f$. 69 | \item\pts{1} Demuestre que el problema tiene existencia y unicidad de soluciones. 70 | \end{itemize} 71 | 72 | \item Considere la ecuación de conservación de masa con una fuente de masa $\theta$: 73 | $$ \parder{\rho}{t} + \dive(\rho \vec v) = \theta. $$ 74 | \begin{itemize} 75 | \item\pts{1} Muestre cómo modificaría la deducción de la ley de conservación de masa para considerar el término fuente de masa $\theta$ y obtener la ecuacion descrita. 76 | \item\pts{1} Suponga un material donde la velocidad de masa $\rho\vec v$ está dada por el inverso del gradiente de densidad: 77 | $$ \rho\vec v = - \ten K \grad \rho, $$ 78 | donde $\ten K$ es un tensor de segundo orden. Muestre que la ecuación resultante es la ecuación del calor para $\rho$ y que la formulación débil para condiciones de Dirichlet homogéneas es: Hallar $\rho(t)$\footnote{Los espacios de funciones tales que están en un espacio de Sobolev en cada instante se llaman espacios de Bochner. Puede ignorar esta dificultad técnica y asumir que es sensato escribir una función $\rho$ tal que para cada instante $t$ se tiene que $\rho(t)$ pertenece a un espacio de Sobolev.} en $H_0^1(\Omega)$ dada una condición inicial $\rho_0$ tal que 79 | $$ \left(\partial_t \rho, v\right) + (\ten K \grad \rho, \grad v) = (\theta, v) \qquad \forall v \in H_0^1(\Omega). $$ 80 | \item\pts{1} Considere una discretización por diferencias finitas implicitas en el tiempo, y muestre cuál es el operador diferencial que aparece en el problema de cada instante $t^n$. Muestre que la formulación débil que obtiene para cada instante $t^n$ es encontrar $\rho^n$ en $H_0^1(\Omega)$ (asumiendo condiciones de Dirichlet homogéneas) tal que 81 | $$ \left(\frac{\rho^n - \rho^{n-1}}{\Delta t}, v\right) + (\ten K \grad \rho^n, \grad v) = (\theta(t^n), v) \qquad \forall v\in H_0^1(\Omega). $$ 82 | Defina claramente todos los objetos matemáticos usados para que esta aproximación esté rigurosamente justificada. Replique el cálculo para una discretización en tiempo dada por el método $\theta$. A problemas como este donde solo una de las variables está discretizada se les conoce como problemas \emph{semi-discretos}. 83 | \item\pts{1} Estudie la invertibilidad del sistema discreto en cada instante $t^n$ para $\theta=0$, $\theta=1/2$, y $\theta=1$. Use el Lema de Lax-Milgram para justificar su respuesta. 84 | \item\pts{1} La condición inf-sup ayuda a caracterizar la sobreyectividad de un operador. Tomando como motivación el ejercicio anterior, explique si es posible establecer una condición inf-sup para el siguiente problema: Encontrar $u$ en $H_0^1(\Omega)$ tal que 85 | $$ (u, v)_0 = \langle f, v\rangle_{H^{-1}\times H_0^1} \qquad \forall v\in H_0^1(\Omega). $$ 86 | Naturalmente, no es posible usar Lax-Milgram ya que el producto en $L^2$ no es elíptico en $H_0^1$. 87 | \item\pts{1} Proponga un esquema de elementos finitos que le permita aproximar el problema semi-discreto a partir de uno completamente discreto. 88 | \end{itemize} 89 | \end{enumerate} 90 | 91 | \todo[inline,color=white!90!black]{\textbf{Nota: } Abriremos un foro en Canvas para revisar cualquier typo y/o error que haya en el enunciado.} 92 | \end{document} 93 | 94 | -------------------------------------------------------------------------------- /tareas/tarea4/main.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \usepackage[utf8]{inputenc} 3 | \usepackage{amsmath, amsthm, amssymb, mathpazo, isomath, mathtools} 4 | \usepackage{subcaption,graphicx,pgfplots} 5 | \usepackage{fullpage} 6 | \usepackage{booktabs} 7 | \usepackage{hyperref} 8 | \usepackage{algorithm, algorithmic} 9 | \usepackage{mathtools} 10 | \usepackage{todonotes} 11 | 12 | \title{Tarea 4} 13 | %\author{Nicol\'as A Barnafi\thanks{Instituto de Ingeniería Biológica y Médica, Pontificia Universidad Católica de Chile, Chile}, Axel Osses\thanks{Departamento de Ingeniería Matemática, Universidad de Chile, Chile}} 14 | %\author{Nicol\'as A Barnafi} 15 | \date{} 16 | 17 | \renewcommand{\vec}{\vectorsym} 18 | \newcommand{\mat}{\matrixsym} 19 | \newcommand{\ten}{\tensorsym} 20 | \DeclareMathOperator{\grad}{\nabla} 21 | \DeclareMathOperator{\dive}{\text{div}} 22 | \DeclareMathOperator{\curl}{\text{curl}} 23 | \DeclareMathOperator{\tr}{\text{tr}} 24 | \DeclareMathOperator{\sym}{\text{sym}} 25 | \newtheorem{remark}{Remark} 26 | \newtheorem{definition}{Definition} 27 | \newcommand{\R}{\mathbb{R}} 28 | \newcommand{\D}{\mathcal{D}} 29 | 30 | \newcommand{\tin}{\text{in}} 31 | \newcommand{\ton}{\text{on}} 32 | 33 | \newtheorem{theorem}{Theorem} 34 | \newtheorem{lemma}{Lemma} 35 | \newcommand{\pts}[1]{[{\bf #1 puntos}] } 36 | 37 | \begin{document} 38 | 39 | \maketitle 40 | \hfill \textbf{Fecha de entrega: 23:59 del 30/05/2025} 41 | 42 | \todo[inline,color=white!90!black]{\textbf{Instrucciones: } La tarea debe ser entregada de manera individual en un informe en formato .pdf a través del buzón habilitado en la plataforma Canvas, donde deben mostrar también el código desarrollado. Para su conveniencia, pueden entregar las tareas en un Jupyter Notebook, de modo que sea más cómodo mostrar el código. En cualquier caso, se debe entregar un único archivo como respuesta a la tarea. La política de atrasos será: se calculará un factor lineal que vale 1 a la hora de entrega y 0 48 horas después. Esto multiplicará su puntaje obtenido. Pueden usar ChatGPT u otros modelos solo a conciencia. El uso de salidas de GPT sin su debida comprensión será severamente sancionado. } 43 | 44 | \begin{enumerate} 45 | 46 | \item\pts{1} Dado un espacio de Hilbert $H$ considere una forma bilineal $a:H\times H\to \R$ acotada y elíptica. Demuestre que también satisface las hipótesis del Lema Generalizado de Lax-Milgram. 47 | 48 | \item\pts{3} Considere el problema de Stokes en $\Omega\subset \R^n$, dado por: Encuentre $\vec u$ en $H_0^1(\Omega,\R^n)$ y $p$ en $L_0^2(\Omega)$ tales que 49 | $$\begin{aligned} 50 | -\mu \Delta \vec u + \grad p &= \vec 0 &&\Omega \\ 51 | \dive \vec u &=0 &&\Omega \\ 52 | \vec u &= \vec 0 && \partial \Omega. 53 | \end{aligned}$$ 54 | Muestre que este problema se puede escribir de forma mixta. Una formulación mixta consiste en encontrar $u$ en $V$ y $p$ en $Q$ tales que, dadas dos formas bilineales $a:V\times V\to \R$ y $b:V\times Q\to \R$ y funcionales lineales $F\in V'$ y $G\in Q'$, se tiene que 55 | $$\begin{aligned} 56 | a(u, v) + b(v, p) &= F(v) &&\forall v\in V\\ 57 | b(u,q) &= G(q) &&\forall q\in Q. 58 | \end{aligned}$$ 59 | Esto se escribe en forma de operadores como 60 | $$ \begin{bmatrix} A & B^T \\ B & 0 \end{bmatrix}\begin{bmatrix}u \\ p \end{bmatrix} = \begin{bmatrix} F \\ G \end{bmatrix}. $$ 61 | Para el problema de Stokes, identifique cuales son los operadores $A$ y $B$, y averigüe por qué es importante considerar 62 | $$L_0^2(\Omega) = \left\{ q \in L^2(\Omega): \int_\Omega q\,dx = 0 \right\}$$ 63 | en lugar de simplemente el espacio $L^2(\Omega)$. 64 | 65 | \item\pts{2} Condiciones suficientes para la existencia de soluciones de un problema mixto son que: 66 | \begin{itemize} 67 | \item $a$ sea una forma acotada y elíptica. 68 | \item $b$ sea continua y tal que satisface la condición inf-sup. 69 | \end{itemize} 70 | Muestre que bajo estas hipótesis, el problema de Stokes en forma monolítica, i.e. escrito en $H = H_0^1(\Omega, \R^n) \times L_0^2(\Omega)$ como 71 | $$ M((u,p), (v,q)) = \ell((v,q)) \qquad\forall (v,q) \in H $$ 72 | donde 73 | $$\begin{aligned} 74 | M((u,p), (v,q)) &:= a(u,v) + b(u,q) + b(v,p) \\ 75 | \ell((v,q)) &:= F(v) + G(q), 76 | \end{aligned}$$ 77 | satisface una condición inf-sup. 78 | 79 | \item Considere el problema ADR, dado por encontrar $u$ en $H_0^1(\Omega)$ tal que 80 | $$ -\mu \Delta u + \vec b \cdot \grad u + cu = f, $$ 81 | para alguna $f$ en $H^{-1}(\Omega)$. 82 | \begin{itemize} 83 | \item\pts{1} Proponga un espacio discreto de elementos finitos que sea conforme en $H^1$ y que permita aproximar a la solución continua del problema ADR. 84 | \item\pts{1} Demuestre que el problema discreto está bien puesto. 85 | \item\pts{2} Demuestre que esta elección de espacio genera una aproximación de la solución que es convergente a la solución. Muestre además la tasa de convergencia teórica esperada para el esquema discreto propuesto. 86 | \end{itemize} 87 | 88 | \item Considere $\Omega = (0,1)^2$. Implemente un código de elementos finitos que aproxime el problema ADR con condición de Dirichlet $u=1$ en el lado derecho y Neumann homogéneo en el resto, puede elegir si aproximar con triángulos o con cuadrados. Para ello: 89 | \begin{itemize} 90 | \item\pts{1} Dado un número de intervalos por lado, implemente las matrices \texttt{IEN} y \texttt{coords}. Muestre su resultado para un cuadrado con dos elementos por lado, y grafique la geometría discretizada con los elementos enumerados para validar el resultado. 91 | \item\pts{3} Considere el elemento de referencia y un elemento global igual al de referencia por simplicidad ($K = \hat K$). Calcule con una regla de cuadratura adecuada la matriz local asociada a cada uno de los operadores del problema y valide los valores numéricos obtenidos con las expresiones analíticas de las integrales: 92 | \begin{itemize} 93 | \item $-\Delta u$ 94 | \item $\vec b \cdot \grad u$ 95 | \item u 96 | \end{itemize} 97 | \item\pts{2} Repita el punto anterior para el vector local con un lado derecho dado por $f(x) = 1$. 98 | \item\pts{3} A partir de las rutinas anteriores, genere una función \texttt{getProblem(mu,b,c,f)} donde $\mu,b,c$ son escalaes y $f:\R\to\R$ que entregue la matriz $\mat A_h$ que define el problema y el lado derecho $\vec F_h$. 99 | \item\pts{2} Explique cómo implementar las condiciones de borde y genere una función que modifique el problema $(\mat A_h, \vec F_h)$ para imponer la condición de Dirichlet dada ($u=1$ en el lado derecho). 100 | \item\pts{3} Usando el método de soluciones manufacturadas, muestre que la solución discreta converge a la continua con la tasa esperada. \emph{Hint: Para calcular el error de aproximación $e_h = u - u_h$, use que la norma se puede dividir por elementos $\|e_h\|_0^2 = \int_\Omega e_h^2\,dx = \sum_e \int_e e_h^2\,dx$ y la integral en cada elemento se puede calcular con cuadratura en el elemento de referencia.} 101 | \end{itemize} 102 | \end{enumerate} 103 | 104 | \todo[inline,color=white!90!black]{\textbf{Nota: } Abriremos un foro en Canvas para revisar cualquier typo y/o error que haya en el enunciado.} 105 | \end{document} 106 | 107 | -------------------------------------------------------------------------------- /tareas/tarea3/main.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \usepackage[utf8]{inputenc} 3 | \usepackage{amsmath, amsthm, amssymb, mathpazo, isomath, mathtools} 4 | \usepackage{subcaption,graphicx,pgfplots} 5 | \usepackage{fullpage} 6 | \usepackage{booktabs} 7 | \usepackage{hyperref} 8 | \usepackage{algorithm, algorithmic} 9 | \usepackage{mathtools} 10 | \usepackage{todonotes} 11 | 12 | \title{Tarea 3} 13 | %\author{Nicol\'as A Barnafi\thanks{Instituto de Ingeniería Biológica y Médica, Pontificia Universidad Católica de Chile, Chile}, Axel Osses\thanks{Departamento de Ingeniería Matemática, Universidad de Chile, Chile}} 14 | %\author{Nicol\'as A Barnafi} 15 | \date{} 16 | 17 | \renewcommand{\vec}{\vectorsym} 18 | \newcommand{\mat}{\matrixsym} 19 | \newcommand{\ten}{\tensorsym} 20 | \DeclareMathOperator{\grad}{\nabla} 21 | \DeclareMathOperator{\dive}{\text{div}} 22 | \DeclareMathOperator{\curl}{\text{curl}} 23 | \DeclareMathOperator{\tr}{\text{tr}} 24 | \DeclareMathOperator{\sym}{\text{sym}} 25 | \newtheorem{remark}{Remark} 26 | \newtheorem{definition}{Definition} 27 | \newcommand{\R}{\mathbb{R}} 28 | \newcommand{\D}{\mathcal{D}} 29 | 30 | \newcommand{\tin}{\text{in}} 31 | \newcommand{\ton}{\text{on}} 32 | 33 | \newtheorem{theorem}{Theorem} 34 | \newtheorem{lemma}{Lemma} 35 | \newcommand{\pts}[1]{[{\bf #1 puntos}] } 36 | 37 | \begin{document} 38 | 39 | \maketitle 40 | \hfill \textbf{Fecha de entrega: 23:59 del 16/05/2025} 41 | 42 | \todo[inline,color=white!90!black]{\textbf{Instrucciones: } La tarea debe ser entregada de manera individual en un informe en formato .pdf a través del buzón habilitado en la plataforma Canvas, donde deben mostrar también el código desarrollado. Para su conveniencia, pueden entregar las tareas en un Jupyter Notebook, de modo que sea más cómodo mostrar el código. En cualquier caso, se debe entregar un único archivo como respuesta a la tarea. La política de atrasos será: se calculará un factor lineal que vale 1 a la hora de entrega y 0 48 horas después. Esto multiplicará su puntaje obtenido. Pueden usar ChatGPT u otros modelos solo a conciencia. El uso de salidas de GPT sin su debida comprensión será severamente sancionado. } 43 | 44 | \begin{enumerate} 45 | 46 | \item Considere $\Omega\subset \R^d$ Lipschitz y acotado, una función $\ten K:\Omega \to \R^{d\times d}$ en $L^\infty(\Omega,\R^{d\times d})$ tal que es simétrica, continua y definida positiva en casi todo punto ($\exists c, C>0: c|\vec x|^2 \leq \vec x^T\ten K\vec x \leq C|\vec x|^2$ c.t.p), una función $\vec b$ tal que $\dive {\vec b}=0$, $\vec b\cdot \vec n\geq 0$ en $\Gamma_N$ y $\vec b\in H(\dive; \Omega)$, una función escalar positiva $a>0$ en $L^\infty(\Omega)$, y un elemento $f$ de $[H^1(\Omega)]'$. En este contexto, considere el problema de Advección-Difusión-Reacción (ADR) 47 | $$ 48 | \begin{aligned} 49 | -\dive \ten K\grad u + \vec b\cdot \grad u + a u &= f &&\text{ en $\Omega$}, \\ 50 | u &= u_D &&\text{en $\Gamma_D$}, \\ 51 | \ten K\grad u\cdot \vec n &= t &&\text{en $\Gamma_N$}, 52 | \end{aligned} 53 | $$ 54 | donde $\partial \Omega=\overline\Gamma_D \cup \overline \Gamma_N$, $u_D$ está en $H^{1/2}(\Gamma_D)$, y $t$ en $H^{-1/2}(\Gamma_N)$. 55 | \begin{enumerate} 56 | \item\pts{2} Encuentre una formulación débil de este problema, definiendo claramente el espacio de soluciones, el espacio de las funciones test y las formas lineales/bilineales involucradas. Para ello, solo debe integrar por partes el operador diferencial de segundo orden. Notar que integrar el término de primer orden generaría nuevos términos de frontera que serían difíciles de analizar. 57 | \item\pts{2} Demuestre que la formulación débil encontrada tiene una solución única usando el Lema de Lax-Milgram. Le será útil demostrar la siguiente identidad: 58 | $$ \int_{\partial\Omega} u^2 (\vec b\cdot n)\,dS = \int_\Omega \dive (u\vec b) u\,dx + \int_\Omega (u\vec b)\cdot \grad u\,dx = \int_\Omega (\vec b\cdot \grad u + u \dive \vec b) u\,dx + \int_\Omega u\vec b\cdot \grad u\,dx $$ 59 | Indique claramente dónde se usan las hipótesis sobre los parámetros. 60 | \end{enumerate} 61 | 62 | \item Considere $\Omega\subset \R^d$ Lipschitz acotado, y el siguiente problema: Hallar $u_1, u_2:\Omega \to \R$ tales que 63 | $$ 64 | \begin{aligned} 65 | -\Delta u_1 + u_2 &= f_1 && \text{en $\Omega$} \\ 66 | -\Delta u_2 - u_1 &= f_2 && \text{en $\Omega$}, 67 | \end{aligned} 68 | $$ 69 | dadas dos funciones $f_1,f_2$. 70 | \begin{enumerate} 71 | \item\pts{2} Muestre, a través de integración por partes de cada ecuación por separado, cuales son las condiciones de borde adecuadas para este problema (Dirichlet, Neumann o una mezcla de ellas). 72 | \item\pts{2} Considere condiciones de borde homogéneas para ambas variables: 73 | $$ u_1 = u_2 = 0 \quad\text{en $\partial\Omega$}. $$ 74 | Escriba la formulación débil del problema, escribiendo claramente cuales son los espacios involucrados, y la regularidad requerida para las funciones $f_1, f_2$. Para esto, le servirá notar que el espacio de soluciones de su problema puede estar dado por $V_0 = H_0^1(\Omega)\times H_0^1(\Omega)$, y por lo tanto $\vec u \in V_0$ si y solo si $\vec u = (u_1, u_2)$, para $u_1, u_2$ en $H_0^1(\Omega)$. 75 | \item\pts{2} Demuestre que el problema tiene una única solución y escriba la cota a-priori de estabilidad que muestra la continuidad de la inversa. 76 | \end{enumerate} 77 | 78 | \item Considere $\mu,\lambda>0$ (parámetros de Lamé), $\Omega\subset\R^2$ Lipschitz acotado. Dado un campo vectorial $\vec u:\Omega\to \R^2$, que llamaremos desplazamiento, definimos su gradiente simétrico como 79 | $$ \ten \varepsilon (\vec u) \coloneqq \frac 1 2\left(\grad \vec u + [\grad \vec u]^T\right), $$ 80 | donde la matriz $\grad \vec u$ está dada puntualmente por $[\grad \vec u]_{ij} = \frac{\partial u_i}{\partial x^j}$. Se define además el tensor de Hooke como el siguiente operador: 81 | $$ \ten \sigma (\vec u) \coloneqq 2 \mu \ten \varepsilon(\vec u) + \lambda \dive (\vec u) \ten I. $$ 82 | Dadas estas definiciones, una fuerza de volumen $\vec f:\Omega\to \R$, y una condición de borde $\vec u_D$, se define el problema de \emph{elasticidad lineal} como 83 | $$ \begin{aligned} 84 | -\dive \ten \sigma (\vec u) &= \vec f &&\text{ en $\Omega$}, \\ 85 | \vec u &= \vec u_d &&\text{en $\Gamma_D$}, \\ 86 | \ten \sigma \vec n &= \vec t &&\text{en $\Gamma_N$}. 87 | \end{aligned} $$ 88 | \begin{enumerate} 89 | \item\pts{1} Demuestre que dada una matriz $A$ simétrica y una matriz $B$ antisimétrica, se tiene que 90 | $$ A : B = 0, $$ 91 | donde $:$ es la contracción de matrices o producto de Frobenius, dado por $X:Y=\sum_{i,j} X_{ij}Y_{ij}$. 92 | \item\pts{1} Demuestre que la fórmula de integración por partes en forma matricial está dada por 93 | $$ \int_{\partial\Omega} \vec v\cdot (\ten \tau \vec n)\,dS = \int_\Omega \dive\ten \tau \cdot \vec v\,dx + \int_\Omega \ten\tau : \grad \vec v\,dx, $$ 94 | donde $\ten \tau:\Omega \to \R^{d\times d}$ pertenece a $\ten H(\dive; \Omega)\coloneqq[H(\dive;\Omega)]^d$, y $\vec v:\Omega \to \R^d$ pertenece a $\vec H^1(\Omega)\coloneqq [H^1(\Omega)]^d$. Notar que el operador $\dive$ aplicado a una función matricial $\ten \tau$ se aplica por filas, es decir que resulta el siguiente vector: 95 | $$ \dive \ten \tau = \begin{bmatrix} \dive \ten \tau_{1,\bullet} \\ \dive \ten \tau_{2,\bullet} \\ \vdots \\ \dive \ten \tau_{d, \bullet} \end{bmatrix} $$ 96 | \item\pts{2} Use los dos resultados anteriores para mostrar que la formulación débil del problema de elasticidad está dada por: Encontrar $\vec u$ en $\vec V_{u_D}$ tal que 97 | $$ \int_\Omega \ten \sigma(\vec u):\ten\varepsilon(\vec v)\,dx = \langle \vec f, \vec v\rangle_{V_0', V_0} + \langle \vec t, \vec v\rangle_{[H^{1/2}(\Gamma_N)]', H^{1/2}(\Gamma_N)} \qquad \forall \vec v \in \vec V_0,$$ 98 | donde 99 | $$ \vec V_{u_D} = \{ \vec v \in \vec H^1(\Omega): \vec v = \vec u_D \text{ en $\Gamma_D$}\} $$ 100 | y 101 | $$ \vec V_0 = \{ \vec v \in \vec H^1(\Omega): \vec v = \vec 0 \text{ en $\Gamma_D$}\}, $$ 102 | con las igualdades en dichas definiciones entendidas en el sentido de las trazas. 103 | \item\pts{1} Muestre que el gradiente simétrico $\varepsilon(\vec u)$ es tal que 104 | $$ \| \ten \varepsilon(\vec u) \|_{0,\Omega} \leq C \| \grad \vec u \|_{0,\Omega} . $$ 105 | \item\pts{3} Demuestre la desigualdad de Körn, dada por 106 | $$ \| \vec u \|_{1,\Omega} \leq C \|\ten \varepsilon(\vec u) \|_{0,\Omega} \qquad \forall \vec u \in [H_0^1(\Omega)]^d. $$ 107 | \emph{Hint: Extienda la demostración de Poincaré para este caso. Notar que la condición $\ten \varepsilon(\vec u) = 0$ implica que $\vec u$ es un movimiento rígido, i.e. una función de la forma $\vec u = \left(\begin{smallmatrix} \alpha x_2 + \beta \\ -\alpha x_1 + \gamma \end{smallmatrix}\right)$, para $\alpha, \beta, \gamma$ constantes arbitrarias. } 108 | \item\pts{2} Usando los resultados anteriores, muestre que el problema de elasticidad lineal tiene una única solución y muestre la cota de estabilidad. 109 | \end{enumerate} 110 | \end{enumerate} 111 | 112 | \todo[inline,color=white!90!black]{\textbf{Nota: } Abriremos un foro en Canvas para revisar cualquier typo y/o error que haya en el enunciado.} 113 | \end{document} 114 | 115 | -------------------------------------------------------------------------------- /tareas/tarea-topicos-1/main.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \usepackage[utf8]{inputenc} 3 | \usepackage{amsmath, amsthm, amssymb, mathpazo, isomath, mathtools} 4 | \usepackage{subcaption,graphicx,pgfplots} 5 | \usepackage{fullpage} 6 | \usepackage{booktabs} 7 | \usepackage{hyperref} 8 | \usepackage{algorithm, algorithmic} 9 | \usepackage{mathtools} 10 | \usepackage{todonotes} 11 | 12 | \newcommand{\example}[1]{\todo[inline,color=green!30!white]{\textbf{Example:} #1}} 13 | 14 | \title{Tarea 1} 15 | %\author{Nicol\'as A Barnafi\thanks{Instituto de Ingeniería Biológica y Médica, Pontificia Universidad Católica de Chile, Chile}, Axel Osses\thanks{Departamento de Ingeniería Matemática, Universidad de Chile, Chile}} 16 | %\author{Nicol\'as A Barnafi} 17 | \date{} 18 | 19 | \renewcommand{\vec}{\vectorsym} 20 | \newcommand{\mat}{\matrixsym} 21 | \newcommand{\ten}{\tensorsym} 22 | \DeclareMathOperator{\grad}{\nabla} 23 | \DeclareMathOperator{\dive}{\text{div}} 24 | \DeclareMathOperator{\curl}{\text{curl}} 25 | \DeclareMathOperator{\tr}{\text{tr}} 26 | \DeclareMathOperator{\sym}{\text{sym}} 27 | \newtheorem{remark}{Remark} 28 | \newtheorem{definition}{Definition} 29 | \newcommand{\R}{\mathbb{R}} 30 | \newcommand{\D}{\mathcal{D}} 31 | 32 | \newcommand{\tin}{\text{in}} 33 | \newcommand{\ton}{\text{on}} 34 | 35 | \newtheorem{theorem}{Theorem} 36 | \newtheorem{lemma}{Lemma} 37 | 38 | \begin{document} 39 | 40 | \maketitle 41 | \hfill \textbf{Fecha de entrega: 23:59 del 09/10/2024} 42 | 43 | \todo[inline,color=white!90!black]{\textbf{Instrucciones: } La tarea debe ser entregada en formato .pdf a través del buzón habilitado en la plataforma Canvas. Luego de la corrección de las tareas, les daré la oportunidad de rehacer todos los ejercicios que tengan errores para poder trabajar sobre sus errores. Por lo mismo, las tareas con atraso tendrán un 1.0 automáticamente. Las preguntas 1,2,3,4 tienen 1 punto. Las preguntas 5,6,10 tienen 2 puntos. Las preguntas 7,8,9 tienen 3 puntos. El puntaje está basado en mi estimación personal de dificultad de las preguntas.} 44 | 45 | \begin{enumerate} 46 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 47 | \item El espacio de movimientos rígidos, dado por el kernel del gradiente simétrico $\varepsilon(u) = \frac 1 2 \left( \grad u + (\grad u)^T \right)$ está dado por el espacio generado por las funciones $(1,0)$, $(0,1)$, y $(-y, x)$. Definimos dicho espacio como $\mathbb{RM}$. Encuentre la proyección a $\mathbb{RM}$ de la functión 48 | $$ f(x,y) = (\sin(x) + \cos(y), \sin(x) - \cos(y)) $$ 49 | en $\mathbb{RM}$ con respecto a los espacios $L^2(\Omega)$ y $H^1(\Omega)$, con $\Omega=[0,1]^2$. 50 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 51 | \item Considere la siguiente función: 52 | $$ f(x) = \begin{cases} 53 | 1 & \text{$x$ in $(0,1)$} \\ 54 | 0 & \text{elsewhere} 55 | \end{cases}. $$ 56 | Muestre que $f$ está en $L^\infty(\R)$ pero no en $W^{1,\infty}(\R)$. Extienda la demostración para alguna función discontinua $\R^2$. 57 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 58 | \item Encuentre una generalización de la desigualdad de Hölder para productos de $n$ funciones: $\| f_1 \hdots f_n \|_{L^1(\Omega)} \leq \| f_1\|_{L^{a_1}(\Omega)} \hdots \| f_n \|_{L^{a_n}(\Omega)}$. Qué condiciones deben cumplir los exponentes $a_i$? 59 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 60 | \item Demuestre las tres identidades de integración por partes mostradas en los apuntes del curso. 61 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 62 | \item (Problemas cuadráticos) Se define la derivada de Gateaux de un funcional $\Pi: V \to \R$ en $u$ y en dirección $v$ como 63 | $$ d\Pi(u)[v] \coloneqq \frac{d}{d\epsilon}\left.\left(\Pi(u + \epsilon v) \right)\right|_{\epsilon=0}. $$ 64 | Esta derivada es una extensión de las derivadas parciales, y coincide con la derivada abstracta (de Frechèt) en el caso en que el funcional resultante sea lineal y continuo. 65 | Muestre que dada una forma bilineal simétrica $a: V\times V \to \R$ y un funcional lineal $L:V\to \R$, se tiene que las condiciones de primer orden del funcional cuadrático $\Pi(u) = \frac 1 2 a(u,u) + L(u)$ corresponden a la ecuación 66 | $$ a(u,v) = L(v) \qquad\forall v \in V. $$ 67 | Muestre además que los puntos $u$ en $V$ que satisfacen dicha ecuación son efectivamente mínimos del funcional cuadrático. Finalmente, muestre que esto caracteriza a las funciones armónicas ($-\Delta u=0$) como aquellas que minimizan la semi norma $H^1$. 68 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 69 | \item (Laplaciano biarmónico) Considere el laplaciano biarmónico $\Delta^2 u$, definido como aplicar dos veces el Laplaciano $\Delta = \sum_i \partial_i^2$. Este operador aparece en el contexto general de la teoría de Kirchhof-Love para modelar la deformación de placas delgadas sometidas a esfuerzos mecánicos. Encuentre la formulación variacional de este problema, y sus correspondientes condiciones de Dirichlet y de Neumann. 70 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 71 | \item (Elasticidad lineal) Considere el tensor de Hooke, tensor del cuarto orden, cuya acción está dada por 72 | $$ \mathbb{C}_\text{Hooke}\tau = \lambda \tr \tau \ten I + 2 \mu \tau, $$ 73 | donde $\lambda, \mu$ se conocen como parámetros de Lamé. Considere además el gradiente simétrico $\varepsilon(u) = \frac 1 2 \left( \grad u + (\grad u)^T \right)$. Encuentre la formulación débil del problema de elasticidad lineal, cuya forma fuerte está dada por 74 | $$ - \dive \mathbb C_\text{Hooke}\varepsilon(\vec u) = \vec f $$ 75 | para alguna fuerza externa $f$. Este problema corresponde a encontrar el desplazamiento de un sólido $\Omega$ sometido a ciertos esfuerzos y/o condiciones de borde, y a una fuerza volumétrica $f$ (como la gravedad). 76 | 77 | Determine la regularidad necesaria de $f$, así como también las condiciones de borde sugeridas por la integración por partes. Finalmente, demuestre que dada una matriz arbitraria $A$ y una simétrica $S$, se tiene que 78 | $$ A : S = \sym(A) : S, $$ 79 | donde $\sym(A) = \frac 1 2\left(A + A^T\right)$, para encontrar una formulación variacional donde la forma bilineal sea simétrica. Para dicha formulación, elija una condición de borde y, para el problema resultante (i) demuestre la existencia y unicidad de soluciones\footnote{El análogo a las desigualdades de Poincaré para este problema se llaman desigualdades de K\"orn, y las puede citar directamente desde el libro de Brenner y Scott. Su mayor dificultad radica en que hay que controlar la norma del gradiente simétrico, no simplemente del gradiente.}, y (ii) proponga un espacio discreto conforme y muestre cuáles son las tasas de convergencia esperadas a partir de la estimación de Céa correspondiente. 80 | 81 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 82 | \item (Condiciones de Dirichlet débiles) Como vimos, las condiciones de borde para un problema como el Laplaciano: 83 | $$ -\Delta u = f \quad\text{en $\Omega$}, \qquad u=g\quad\text{en $\partial\Omega$} $$ 84 | deben ser tratadas con cuidado. Una estrategia más robusta para tratar esta dificultad es la de imponer la condición de Dirichlet de manera débil. Este será nuestro primer ejemplo de un método \emph{no-conforme}. Para esto, revise la formulación por penalización y el método de Nitsche, descritos en el siguiente trabajo: 85 | 86 | $$ \text{https://doi.org/10.1007/s10013-024-00702-1}. $$ 87 | 88 | Para el análisis, será útil revisar el siguiente paper: 89 | 90 | $$ \text{https://doi.org/10.1016/0377-0427(95)00057-7} $$ 91 | 92 | Para esta pregunta, en ambas formulaciones (penalización y Nitsche): (i) Encuentre la formulación variacional discreta, (ii) Demuestre existencia y unicidad de la formulación discreta (encontrará la norma correcta para usar en las referencias entregadas), (iii) encuentre un esquema de elementos finitos convergentes y demuestre la tasa de convergencia, y (iv) calcule las tasas de convergencia numéricas para alguna solución manufacturada conveniente. Responda además la siguiente pregunta: Qué argumentos a favor y en contra hay de cada formulación? 93 | 94 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 95 | \item (Problema de Helmholtz) Vimos que el problema de Helmholtz no es elíptico: Encontrar $u$ en $H^1(\Omega)$ tal que 96 | $$ 97 | \left\{\begin{aligned} 98 | -\Delta u - k^2 u&= 0 &&\Omega \\ 99 | u &= g &&\partial\Omega 100 | \end{aligned}\right. 101 | $$ 102 | La ecuación de Helmholtz se obtiene típicamente al hacer la transformada de Fourier en tiempo de la ecuación de onda. Así, desaparece la dependencia en tiempo del problema. Típicamente esta ecuación, por la misma razón, se formula en el plano complejo, así que la ecuación que estamos estudiando corresponde solo a la parte real de la solución. 103 | 104 | Utilice la estimación de Céa vista en clases para encontrar un esquema de elementos finitos convergente para este problema. Calcule numéricamente las tasas de convergencia para alguna solución manufacturada, y muestre que la convergencia se tiene solo para una malla lo suficientemente fina. 105 | 106 | 107 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 108 | \item (Problema de advección-difusión) Muestre que, para una malla lo suficientemente fina, y dadas funciones $f$ en $H^{-1}(\Omega)$ y $\vec b$ en $\vec L^\infty(\Omega)$, el problema de advección-difusión: 109 | $$ 110 | \left\{\begin{aligned} 111 | -\Delta u + \vec b\cdot \grad u &= \vec f &&\Omega \\ 112 | u &= 0 &&\partial\Omega 113 | \end{aligned}\right. 114 | $$ 115 | (i) posee existencia de soluciones, (ii) existe un esquema de elementos finitos conformes para aproximarlo, y (iii) muestre las tasas de convergencia de dicho esquema. \textbf{Nota: }Probablemente, el efecto de "malla suficientemente fina" se note solo si $\dive \vec b\neq 0$, ya que en ese caso el problema es elíptico. 116 | 117 | \end{enumerate} 118 | 119 | \todo[inline,color=white!90!black]{\textbf{Nota: } No he desarrollado los problemas, así que probablemente tengan typos y/o errores. Podemos discutir esto cuando quieran, o me pueden notificar por correo en \texttt{nicolas.barnafi@uc.cl}.} 120 | \end{document} 121 | 122 | -------------------------------------------------------------------------------- /tareas/projects/main.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \usepackage[utf8]{inputenc} 3 | \usepackage{amsmath, amsthm, amssymb, mathpazo, isomath, mathtools} 4 | \usepackage{subcaption,graphicx,pgfplots} 5 | \usepackage{fullpage} 6 | \usepackage{booktabs} 7 | \usepackage{hyperref} 8 | \usepackage{algorithm, algorithmic} 9 | \usepackage{mathtools} 10 | \usepackage{todonotes} 11 | 12 | \title{Course Projects} 13 | %\author{Nicol\'as A Barnafi\thanks{Instituto de Ingeniería Biológica y Médica, Pontificia Universidad Católica de Chile, Chile}, Axel Osses\thanks{Departamento de Ingeniería Matemática, Universidad de Chile, Chile}} 14 | %\author{Nicol\'as A Barnafi} 15 | \date{} 16 | 17 | \renewcommand{\vec}{\vectorsym} 18 | \newcommand{\mat}{\matrixsym} 19 | \newcommand{\ten}{\tensorsym} 20 | \DeclareMathOperator{\grad}{\nabla} 21 | \DeclareMathOperator{\dive}{\text{div}} 22 | \DeclareMathOperator{\curl}{\text{curl}} 23 | \DeclareMathOperator{\tr}{\text{tr}} 24 | \DeclareMathOperator{\sym}{\text{sym}} 25 | \newtheorem{remark}{Remark} 26 | \newtheorem{definition}{Definition} 27 | \newcommand{\R}{\mathbb{R}} 28 | \newcommand{\D}{\mathcal{D}} 29 | \newcommand{\parder}[2]{\frac{\partial\,#1}{\partial\,#2}} 30 | 31 | \newcommand{\tin}{\text{in}} 32 | \newcommand{\ton}{\text{on}} 33 | 34 | \newtheorem{theorem}{Theorem} 35 | \newtheorem{lemma}{Lemma} 36 | \newcommand{\pts}[1]{[{\bf #1 puntos}] } 37 | 38 | \begin{document} 39 | 40 | \maketitle 41 | Course projects must be developed in groups of 1--3 people. Presentation dates can be flexible but will be preferrably taking place during the last week of lessons of the semester. 42 | 43 | \begin{enumerate} 44 | \item \textbf{Heat equation analysis.} This equations consists in the following: Find $u$ in a Bochner space $L^2(0,T;H_{\Gamma_D}^1(\Omega))$ such that for a heat source $f$ such that $f(t)$ in $[H_{\Gamma_D}^1(\Omega)]'$ and an initial condition $u_0$ in $L^2(\Omega)$ it holds that 45 | $$ \parder{u}{t} - \Delta u = f \qquad\text{in $(0,T]\times \Omega$}. $$ 46 | This project's goals are: 47 | \begin{itemize} 48 | \item To study the well-posedness theory of parabolic equations in a Bochner space setting. 49 | \item To study and analyze its spatial discretization with FEM. 50 | \item To study and analyze its time discretization using the $\theta$-method. 51 | \item To validate numerically the computed convergence rates. 52 | \end{itemize} 53 | 54 | The following references might be useful: 55 | \begin{itemize} 56 | \item Quarteroni, Valli. Numerical approximation of partial differential equations, 1994. 57 | \item Thomée. Galerkin finite element methods for parabolic problems, 2007. 58 | \item Evans. Partial Differential Equations, 2022. 59 | \end{itemize} 60 | 61 | \item \textbf{The Helmholtz problem.} This problem consists in finding the eigenvalues of the Laplace operator, i.e. a pair $(\omega, u)$ such that 62 | $$ -\Delta u = \omega u. $$ 63 | For this project you will have to 64 | \begin{itemize} 65 | \item Study the well-posedness of the problem using Fredholm theory and G\r arding inequalities. 66 | \item Study the spectral properties of the Laplacian for Dirichlet, Neumann, and mixed boundary conditions. 67 | \item Study the spectral properties of the discretized Laplacian under Dirichlet, Neumann, and mixed boundary conditions. 68 | \item Study the convergence of the eigenvalues and eigenvectors and devise strategies to matche the discrete eigenvalues to the corresponding continuous ones. 69 | \item Validate numerically the convergence rates. 70 | \end{itemize} 71 | The following references will be useful for this project: 72 | \begin{itemize} 73 | \item Boffi, Finite element approximation of eigenvalue problems. 2010. 74 | \item Babuska, Osborne. Eigenvalue problems in \emph{Handbook of Numerical Analysis}. 1990. 75 | \end{itemize} 76 | \item \textbf{The Darcy problem.} This is a mixed problem, given by: Find $u$ in $\vec H(\dive)$ and $p$ in $L^2(\Omega)$ such that for some $f$ in $L^2(\Omega)$ it holds that 77 | $$ \begin{aligned} 78 | u + \grad p &= 0 && \Omega \\ 79 | \dive u &= f && \Omega \\ 80 | u\cdot \vec n &= 0 && \Gamma_D \\ 81 | p &= p_0 && \Gamma_N. 82 | \end{aligned} $$ 83 | For this project you will have to 84 | \begin{itemize} 85 | \item Study how this problem is related to the Poisson problem, and when they are equivalent. 86 | \item Study the LBB theory and show that it applies to it 87 | \item Study the discrete problem and show that the LBB conditions hold 88 | \item Validate numerically the convergence rates 89 | \end{itemize} 90 | \item \textbf{The steady Navier-Stokes equation.} The steady Navier-Stokes equation consists in finding a velocity field $\vec u$ in $H_D^1(\Omega, \R^d)$ and a pressulre $p$ in $L^2(\Omega)$ such that 91 | $$\begin{aligned} 92 | -\mu \Delta \vec u + [\grad \vec u] \vec u + \grad p &= \vec f && \Omega \\ 93 | \dive \vec u &= 0 && \Omega \\ 94 | \vec u = \vec u_D && \Gamma_D \\ 95 | \ten \sigma(\vec u, p)\vec n &= \vec t && \Gamma_N, 96 | \end{aligned}$$ 97 | where $\vec f$ in $[H_d^1(\Omega, \R^d)]'$ is a given load, $\vec u_D$ in $H^{1/2}(\Gamma_D)$ is a Dirichlet boundary condition, $\ten\sigma(\vec u, p) = \mu\grad \vec u - p\ten I$ is the stress tensor, $\vec n$ is the normal vector, and $\vec t$ in $H^{-1/2}(\Gamma_N)$ is a given surface traction. The objective of this project is to investigate the techniques required to handle the nonlinearity of this problem in order to guarantee the existence (and possibly uniqueness) of solutions. This implies: 98 | \begin{itemize} 99 | \item To understand the saddle point structure of the problem 100 | \item To formulate an auxiliary linear problem that defines a fixed-point operator 101 | \item To establish conditions that guarantee contractiveness 102 | \item To study extensions using Schauder and Brower fixed-point theorems 103 | \item To use the fixed-point operator to devise a solution strategy of the nonlinear problem 104 | \item To use such techniques to establish the convergence of a FEM scheme 105 | \item To validate all theoretical claims numerically 106 | \end{itemize} 107 | \item \textbf{The unsteady Navier-Stokes equation.} Adding a time derivative to the momentum equation in the previous equation yields the unsteady NS equation. The scope of this project is to study the numerical instabilities that this equation presents, and some of the remedies that can be used. In particular, provide stabilization strategies for 108 | \begin{itemize} 109 | \item Inf-sup instabilities 110 | \item Reynolds instabilities 111 | \item Turbulence modeling 112 | \end{itemize} 113 | All tests shall be performed on a standard flow pasta a cylinder test (in 2D) to obtain physically relevant solutions to be compared. 114 | \item \textbf{The incompressible elasticity equation.} The incompressible elasticity equation is given by 115 | $$\begin{aligned} 116 | -\dive\left( \ten P - \lambda J \ten F^{-T}\right) &= \vec f &&\Omega, \\ 117 | J&= 1 &&\Omega, 118 | \end{aligned}$$ 119 | plus boundary conditions. The idea of this project is to correctly employ a Lagrange multiplier procedure to derive this problem for hyperelastic materials, and then use homotopy continuation to study certain bifurcation phenomena present in nonlinear elasticity. For this: 120 | \begin{itemize} 121 | \item To correctly derive the model 122 | \item To formulate a model with rotating boundary conditions (Dirichlet vs Neumann approaches) 123 | \item To devise a parameter to be used for the continuation 124 | \item To push the model until bifurcation happens 125 | \item To study polyconvexity and use materials that satisfy such a condition (and not convexity) 126 | \item To revise (not in detail) bifurcation theory to justify the observations 127 | \item What can be said about approximation properties? 128 | \end{itemize} 129 | \item \textbf{The Keller-Segel equation.} Chemotaxis describes the dynamics of species that are on one hand attracted and on the other one repelled by certain species. This is described by the following system of equations: 130 | $$\begin{aligned} 131 | \partial_t u &= \dive\left(D_1(u,c)\grad u - \chi(u,c) \grad c\right) && \Omega \\ 132 | \partial_t c &= D_2\Delta c - g(c) c + f(c) u && \Omega \\ 133 | [D_1(u,c)\grad u]\cdot \vec n = \grad c\cdot \vec n &= 0 && \partial \Omega, 134 | \end{aligned} $$ 135 | with some initial conditions for $u,c$. The idea of this project is 136 | \begin{itemize} 137 | \item To propose a time discretization scheme for the model 138 | \item To study the well-posedness of the time-discrete problem 139 | \item To establish the convergence of a FEM scheme for the semi-discrete problems 140 | \item To study the concept of Turing instabilities that give rise to pattern formation 141 | \item To simulate scenarios that are Turing stable and unstable. 142 | \end{itemize} 143 | 144 | \item \textbf{An optimal control problem.} An optimal control problem is a PDE-constrained optimization problem. The idea of this project is to study the discretization of a simple optimal control problem, given by the volume control of the Laplacian. This can be stated as the following minimization problem: 145 | $$ \begin{aligned} \min_{u,\mu} && \frac 1 2 \|u - u_\Omega\|_0^2 &+ \alpha \|\mu\|_0^2 && \\ 146 | \text{s.t.} && -\Delta u &= \mu && \Omega \\ 147 | && u &= 0 && \partial\Omega 148 | \end{aligned} $$ 149 | This project consists in: 150 | \begin{itemize} 151 | \item To study the adjoint method 152 | \item Establishing the well-posedness of this problem using the Stampacchia theorem 153 | \item To formulate a FEM approximation of the problem using either optimize-then-discretize or discretize-then-optimize strategies 154 | \item Show that optimization and discretization in this setting \emph{do not commute} 155 | \item Show the convergence rates of the model 156 | \item To implement the problem and validate all theoretical claims numerically 157 | \end{itemize} 158 | 159 | For this problem, the following references will be useful: 160 | \begin{itemize} 161 | \item Tröltzsch, Optimal control of partial differential equations: theory, methods, and applications. 2010. 162 | \item Manzoni, Quarteroni, Salsa. Optimal control of partial differential equations. 2021. 163 | \end{itemize} 164 | 165 | \item \textbf{Wave equation.} The scope of this project is that of studying the wave equation, given by finding a function $u(t)$ in $H_0^1(\R^d)=\{v\in H^1(\R^d): \lim_{|x|\to \infty} v(x) = 0\}$ such that 166 | $$ c\parder{^2 u}{t^2} -\Delta u=0 $$ 167 | for some initial condition. The scope of this project is to: 168 | \begin{itemize} 169 | \item Study the formulation of symplectic time integration methods that guarantee energy conservation 170 | \item Study the formulation of boundary conditions that allow for a truncation of the domain (unbounded into a bounded one) 171 | \item Implement different symplectic time integrators and study the well-posedness of the semi-discrete problem 172 | \item Study the convergence of a FEM scheme for the semi-discrete problem and establish its convergence properties 173 | \item Validate all theoretical propositions numerically 174 | \end{itemize} 175 | 176 | \paragraph{\textbf{Flexibilities. }} All projects will be assumed to be developed in high level software such as Firedrake/NGSolve/FEniCS/etc, so that the implementation of a FEM code is not part of the difficulty. Still, flexilibity with respect to the stated goals will be considered for projects if you want to explore different topics, such as: 177 | \begin{itemize} 178 | \item Implementation in a lower-level library, such as \texttt{deal.ii} [C++], \texttt{PETSc} [C], Ferrite [Julia]. 179 | \item Study of preconditioning strategies using spectrally equivalent operators. 180 | \item Study of parallel implementations. 181 | \item Study of iterative methods such as Uzawa schemes, or any infinite-dimensional optimization problem. 182 | \item Others you can think of. 183 | \end{itemize} 184 | 185 | \paragraph{\textbf{Inflexibilities.}} A final grade in the project below a 50\% means failing the entire course, and attendance to the final presentation is mandatory for all group members unless timely justified. Each projects can be taken by up to two groups. It will be fundamental for all projects to have 186 | \begin{enumerate} 187 | \item A derivation of the model from fundamental principles 188 | \item Continuous and discrete analysis 189 | \item A convergence analysis showing that there is a controlable gap between the discrete solutions and the continuous ones 190 | \item Two simulations: a numerical validation of the computed convergence rates and a physically significant one 191 | \end{enumerate} 192 | \end{enumerate} 193 | 194 | \end{document} 195 | 196 | -------------------------------------------------------------------------------- /chapters/beyond-linearity.tex: -------------------------------------------------------------------------------- 1 | The theory of well-posedness for linear problems, centered on the Lax-Milgram lemma~\ref{lemma:lax-milgram} and then generalized to the inf-sup condition, provides a robust framework for proving existence and uniqueness of solutions. Many models, however, are inherently nonlinear. Consider, for instance, a semilinear problem of the form 2 | \begin{equation*} 3 | \begin{aligned} 4 | -\Delta u + F(u) &= f &&\tin \Omega, \\ 5 | u &= 0 && \ton \partial\Omega. 6 | \end{aligned} 7 | \end{equation*} 8 | Its weak formulation, $(\nabla u, \nabla v) + (F(u),v) = (f,v)$, results in a form $a(u;v)$ that depends nonlinearly on $u$, and consequently, the Lax-Milgram lemma is not applicable. Thus, we need a new set of tools to establish the well-posedness of such problems. 9 | 10 | This chapter introduces the foundational methods of nonlinear functional analysis. Unlike in the linear setting, there is no single, universally applicable theorem for existence and uniqueness. Instead, we will build a toolkit of three distinct and powerful approaches, each suited for a different class of nonlinear problems. 11 | 12 | We will begin with a \emph{local analysis} based on the inverse function theorem, which provides conditions for a unique solution to exist in a neighborhood of a known solution. We then move to more global methods founded on the \emph{fixed point theorems} of Banach, Brouwer, and Schauder, which provide existence results under assumptions of contractivity or compactness. Finally, we will explore the theory of \emph{monotone operators}, a nonlinear generalization of positive definite operators, and establish the Minty-Browder theorem, a cornerstone result that guarantees the existence of solutions for coercive and monotone operators. 13 | 14 | \section{Local analysis}\label{sec:local-analysis} 15 | Consider the set of \emph{invertible linear operators} 16 | \begin{equation} 17 | \inv(X,Y) \coloneqq \left\{ A\in \mathcal{L}(X,Y): A \text{ is invertible} \right\}. 18 | \end{equation} 19 | For generic $C^1$ functions, we have a local inversion result, known as the \emph{inverse function theorem}. 20 | \begin{theorem}[Inverse function theorem]\label{thm:local-invertibility} 21 | Consider $F\in C^1(X,Y)$, $F'(u^*)\in\inv(X,Y)$. Then, $F$ is \emph{locally invertible} at $u^*$, i.e. 22 | \begin{equation} 23 | dF^{-1}(v) = (F'(u))^{-1}, \quad u=F^{-1}(v), 24 | \end{equation} 25 | and $F^{-1}\in C^1$. 26 | \end{theorem} 27 | 28 | \example{ 29 | Consider the nonlinear problem 30 | \begin{equation*} 31 | \begin{aligned} 32 | \Delta u + u^p &= h, &&\tin \Omega,\\ 33 | u&=0, &&\ton\partial\Omega, 34 | \end{aligned} 35 | \end{equation*} 36 | where $p>1$, $h\in H^{-1}(\Omega)$, so $F:X\to Y$ with $X\coloneqq H_0^1(\Omega)$ and $Y\coloneqq H^{-1}(\Omega)$. We compute 37 | \begin{equation} 38 | dF(u)[w] = -\Delta w + p u^{p-1} w. 39 | \end{equation} 40 | We want to show that $dF(u)\in\inv(X,Y)$. That is, for a given $g\in Y$, we need to prove that there exists a $w\in X$ that solves 41 | \begin{equation*} 42 | a(w,v) = (\nabla w, \nabla v) + p(u^{p-1}w,v) = \langle g,v \rangle \quad \forall v\in X. 43 | \end{equation*} 44 | We have proven many times before the bound for the first term, so let us focus in the second: 45 | \begin{equation*} 46 | |(u^{p-1}w,v)|\leq \|u^{p-1} \|_{L^\infty} \|w\|_{0} \|v \|_{0}. 47 | \end{equation*} 48 | Note that if $u\geq 0$, then $a$ is elliptic. Then, if $u\in X\cap L^\infty \cap \{u\geq 0\}$ we have that $dF(u)\in\inv(X,Y)$ and we can use Theorem~\ref{thm:local-invertibility}. 49 | } 50 | 51 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 52 | \section{Fixed point theorems}\label{sec:fixed-point-theorems} 53 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 54 | The main idea is that certain nonlinearities would be less terrible if we could fix one of the functions. For example, consider the problem of finding a function $u$ such that 55 | \begin{equation} 56 | \begin{aligned} 57 | -\Delta u &= \sin (u) &&\tin \Omega, \\ 58 | u &= 0 && \ton \partial\Omega. 59 | \end{aligned} 60 | \end{equation} 61 | This problem is non-linear, but given $w$, finding $u$ such that 62 | \begin{equation}\label{eq:fixed-point-sin} 63 | -\Delta u = \sin(w) 64 | \end{equation} 65 | is linear on $u$. It also induces a mapping $w \mapsto T(w) = u$ such that, \emph{if it has a fixed point} $u=T(u)$, this fixed point solves the initial problem. For this section, we follow~\cite{ciarlet2013linear,pata2019fixed}. 66 | \begin{definition}[Non-expanding and contractive maps]\label{def:nonexpanding-contractive-maps} 67 | A Lipschitz function $f:X\to X$ 68 | \begin{equation*} 69 | ||f(x)-f(y)||\leq L||x-y|| 70 | \end{equation*} 71 | is said to be \emph{non-expansive} if $L=1$, and a \emph{contraction} if $L<1$. 72 | \end{definition} 73 | \begin{theorem}[Banach fixed point]\label{thm:banach-fixed-point} 74 | Let $X$ be a complete metric space and $f$ a contraction of constant $\lambda<1$. Then $f$ has a unique fixed point. 75 | \begin{proof} 76 | Set $x^{n+1}=f(x^n)$, which we typically refer to as a \emph{Picard iteration}, and consider some initial $x^0\in X$. By induction we get that 77 | \begin{tightalign*} 78 | \|x^{n+1}-x^n\| &\leq \lambda \|x^n-x^{n-1}\|\leq \underbrace{\dots}_\text{induction} \lambda^n\|x^1-x^0\|\\ 79 | \Rightarrow \| x^{n+m}-x^n\| &\leq \|x^{n+m}-x^{n+m-1}\|+ \cdots = \|x^{n+1}-x^n\|\\ 80 | &\leq (\lambda ^{n+m-1}+\cdots+\lambda^n)\|x^1-x^0\|\\ 81 | &=\lambda^n\left(\displaystyle\sum_{j=0}^{m-1}\lambda^j\right)\|x^1-x^0\|\\ 82 | &\leq\lambda^n\left(\displaystyle\sum_{j=0}^\infty\lambda^j\right)\|x^1-x^0\|\\ 83 | &=\lambda^n\dfrac{1}{1-\lambda}\|x^1-x^0\| 84 | \end{tightalign*} 85 | which implies that $\{x_n\}_n$ is Cauchy, and since $X$ is complete, this gives $x_n\to \bar{x}$. Finally, since $f$ is continuous, $f(\bar{x}) = \lim_{n\to \infty}f(x_n) = \lim_{x\to\infty}x_{n+1}=\bar{x}$. 86 | \end{proof} 87 | \end{theorem} 88 | We note that if $X$ is compact, then this theorem can be extended to \emph{weak} contractions, i.e. $\|f(x)-f(y)\|< \|x-y\|$.\\ 89 | 90 | \example{ 91 | Consider problem~\eqref{eq:fixed-point-sin}, where we note that the Lax-Milgram lemma~\eqref{lemma:lax-milgram} immediately yields the existence of a unique solution $u$ for each $w$. From the mean value theorem~\eqref{thm:mean-value-banach} one has that differentiable functions satisfy $f(x) - f(y) = f'(\xi)(x-y)$ for some $\xi$ in $(x,y)$. This in particular shows that $\sin$ is a Lipschitz function as it is differentiable, with $\sin' = \cos$, and $\sup_{\xi\in \R}\cos(\xi) = 1$. Using this fact we can compute the following for some given $w_1, w_2$: 92 | \begin{equation*} 93 | -\Delta(T(w_1)-T(w_2))= \sin(w_1)-\sin(w_2), 94 | \end{equation*} 95 | and the a priori estimate gives 96 | \begin{equation*} 97 | \|T(w_1)-T(w_2)\|\leq C\|\sin(w_1)-\sin(w_2)\|\leq C \|w_1-w_2\|. 98 | \end{equation*} 99 | Then, if $C<1$ there exists $\bar u$ such that $T(\bar u)= \bar u$. 100 | } 101 | \begin{theorem}[Brouwer fixed point]\label{thm:brouwer-fixed-point} 102 | Set $K$ a non-empty, compact and convex subset of a finite-dimensional Banach space. Then, every continuous function $f:K\to K$ has a fixed point. 103 | \end{theorem} 104 | \example{ 105 | Let's go back to~\eqref{eq:fixed-point-sin}, and consider its Galerkin scheme for some given $w_h$: 106 | \begin{equation*} 107 | (\grad u_h,\grad v_h)=(\sin(w_h),v_h) \qquad \forall v_h \in V_h 108 | \end{equation*} 109 | where $\dim(V_h)<\infty$. Then, from the a priori bound we have 110 | \begin{equation*} 111 | \|u_h\|_{1}\leq C||\sin (w_h)||< C 112 | \end{equation*} 113 | using that $|\sin (x)|\leq 1$. This implies that $u_h$ is contained in a finite-dimensional ball of $V_h$, $\mathcal{B}(0,r), r=C$. To show that it is continuous, we consider two functions $w_1,w_2$ in $V_h$, and obtain the difference equation setting $u_i=T(w_i)$: 114 | \begin{equation*} 115 | (\grad [u_1 - u_2], \grad v_h) = (\sin(w_1) - \sin(w_2), v_h) \qquad \forall v_h \in V_h. 116 | \end{equation*} 117 | Again, using the a priori bound we use $\sin'(x) = \cos(x) \leq 1$ to obtain 118 | \begin{equation*} 119 | \| u_1 - u_2\|_1 \leq \|w_1 - w_2\|_0. 120 | \end{equation*} 121 | This yields that $T_h:\mathcal{B}(0,r)\to \mathcal{B}(0,r)$ is continuous and thus has at least one fixed point. 122 | } 123 | 124 | From the previous result we can extend to $h\to 0$ through compactness. This generalization is known as Schauder's fixed point theorem, which has two forms. 125 | \begin{theorem}[Schauder fixed point]\label{thm:schauder-fixed-point} 126 | \begin{enumerate} 127 | \item Let $K$ be a compact, convex subset of a normed space $X$, and $f:K\to K$ a continuous mapping. Then, $f$ has at least one fixed point. 128 | \item Let $\mathcal{C}$ a closed, convex subset of a Banach space $X$, and $f:\mathcal{C}\to \mathcal{C}$ continuous s.t. $\overline{f(\mathcal{C})}$ is compact. Then $f$ has at least one fixed point. 129 | \end{enumerate} 130 | \end{theorem} 131 | \example{ 132 | Consider again~\eqref{eq:fixed-point-sin}, with its induced Picard mapping $w \mapsto u\coloneqq T(w)$ given by 133 | \begin{equation} 134 | \begin{aligned} 135 | -\Delta u &= \sin(w) && \tin \Omega,\\ 136 | u&=0 &&\ton \partial\Omega, 137 | \end{aligned} 138 | \end{equation} 139 | whose solution is guaranteed by Lax-Milgram's lemma, with $u$ in $H_0^1(\Omega)$. From the a priori bound $\|u||\leq\dfrac{1}{\alpha}\|f\|$ we get $\|u\|_1\leq C\|\sin(w)\|_0\leq C|\Omega|=:r_0$. Our candidate set will be $\mathcal C = \bar{B}(0,r_0)\subset L^2(\Omega)$ which is closed, convex and contained in both $L^2(\Omega)$ and $H_0^1(\Omega)$. To prove that $\overline{T(\mathcal C)}$ is compact, consider a sequence $(w_k)_{k>1}$ in $\mathcal C$, which yields $(v_k)_{k>1}=(T(w_k))_{k>1}$ in $\mathcal C$ and additionally $v_k\in H_0^1(\Omega)$ for all $k$. Now, we note that we have a sequence in $H^1$, and as the unit ball is weakly compact, we obtain that is has some weakly convergent sequence in $H^1$. The final detail is that, if we consider $\mathcal C$ to be a ball in $L^2$, then the mapping $T$ can be written additionally as $T\circ i$, where $i:H^1(\Omega)\hookrightarrow L^2(\Omega)$ is the compact embedding of $H^1$ into $L^2$. We thus have that $T\circ i$ is the composition of a continuous and a compact mapping, and thus a compact map. This in particular implies that our sequence $(v_k)$ has a strongly convergent subsequence in $L^2$. This concludes that $T:L^2(\Omega)\to L^2(\Omega)$ is compact, and thus we can use Schauder's fixed point theorem to show the existence of a fixed point $\bar u = T(\bar u)$ in $L^2(\Omega)$. Naturally, the equation itself reveals that actually $\bar u$ belongs to $H_0^1(\Omega)$, so we have some additional regularity on the fixed point we just found. 140 | } 141 | A consequence of the Schauder fixed point theorem is the Schaefer fixed point theorem. We state it for completeness without proof. 142 | \begin{theorem}[Schaefer fixed point]\label{thm:schaefer-fixed-point} 143 | Let $X$ be a Banach space and $f:X\to X$ a compact operator. If 144 | \begin{equation} 145 | \{x\in X: \sigma f(x)=x,\sigma\in[0,1]\}\subset B(0,r) 146 | \end{equation} 147 | for some $r>0$, then there exists a fixed point of $f$. 148 | \end{theorem} 149 | 150 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 151 | 152 | \section{Monotone operators}\label{sec:monotone} 153 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 154 | Among nonlinear operators, there is a class of operators that provide efficient means to prove existence and uniqueness of solutions. These operators are called \emph{monotone}, and they have been extensively studied. We follow this section closely from~\cite{ciarlet2013linear}. 155 | \begin{definition}[Monotone operator]\label{def:monotone-operator} 156 | Let $V$ be a normed vector space and $\langle\cdot,\cdot\rangle$ its duality pairing. An operator $A:V\to V'$ is called \emph{monotone} if 157 | \begin{equation} 158 | \langle A(v)-A(u),v-u\rangle \geq 0 \qquad \forall u,v\in V, 159 | \end{equation} 160 | and \emph{strictly monotone} if the inequality is strict for $u\neq v$. 161 | \end{definition} 162 | We note that if $f:X\to \R$ is a convex and differentiable function, then $\nabla f$ is a monotone operator. Indeed, from convexity applied to $x,y\in X$ and adding we get 163 | \begin{tightalign*} 164 | f(x) &\geq f(y) + \langle \nabla f(y), x-y\rangle\\ 165 | f(y) &\geq f(x) + \langle \nabla f(x), y-x\rangle\\ 166 | \implies 0 &\geq \langle \nabla f(y)-\nabla f(x),x-y\rangle \\ 167 | \implies 0&\leq \langle \nabla f(x)-\nabla f(y),x-y\rangle. 168 | \end{tightalign*} 169 | We can also build monotone operators from an elliptic operator. If $A:V\to V'$ is an operator such that $a(u,v) = \langle A(u),v\rangle$ is continuous (linear) and elliptic, then $A$ is strictly monotone: 170 | \begin{tightalign*} 171 | \langle A(u)-A(v),u-v\rangle &= \langle A(u-v), u-v\rangle\\ 172 | &= a(u-v,u-v)\\ 173 | &\geq \alpha \|u-v\|^2 \geq 0, 174 | \end{tightalign*} 175 | and the inequality is strict for $u\neq v$ since $\|\cdot\|$ is a norm. 176 | 177 | A very useful property can be derived from the monotonicity in the case that $V$ is complete. 178 | \begin{theorem}\label{thm:lipschitz-property-monotone} 179 | If $V$ is a real Banach space and $A:V\to V'$ is monotone, then $A$ is locally bounded, i.e. for any $u\in V$ there exists $r=r(u)>0$ and $\rho=\rho(u)>0$ such that 180 | \begin{equation*} 181 | \|u-v\|\leq r \implies \|A(u)-A(v)\|\leq \rho, 182 | \end{equation*} 183 | which similar to a Lipschitz property. Moreover, if $A$ is linear, then $A$ is continuous. 184 | \end{theorem} 185 | 186 | We now introduce the notion of coercive ($\neq$ elliptic) and hemicontinuous operators. 187 | \begin{definition}[Coercive operator]\label{def:coercive-operator} 188 | An operator $A:V\to V'$ is \emph{coercive} if 189 | \begin{equation*} 190 | \lim_{\|v\|\to \infty} \frac{\langle A(v),v\rangle}{\|v\|} = +\infty. 191 | \end{equation*} 192 | \end{definition} 193 | \begin{definition}[Hemicontinuous operator]\label{def:hemicontinuous-operator} 194 | An operator $A:V\to V'$ is \emph{hemicontinuous} if for every $u,v,w\in V$ there exists $t_0=t_0(u,v,w)>0$ such that the map 195 | \begin{tightalign*} 196 | \varphi:(-t_0,t_0)&\to \R\\ 197 | t&\mapsto \langle A(u+tv),w\rangle 198 | \end{tightalign*} 199 | is continuous at $t=0$. 200 | \end{definition} 201 | Hemicontinuity is a weaker property than continuity, as we shall prove in the following theorem. 202 | \begin{theorem} 203 | If $A:V\to V'$ is continuous, then it is hemicontinuous. 204 | \begin{proof} 205 | Let $\varphi(t)=\langle A(u+tv),w\rangle$. Then, we get 206 | \begin{tightalign*} 207 | |\varphi(t)-\varphi(0)| &= |\langle A(u+tv),w\rangle - \langle A(u),w\rangle|\\ 208 | &\leq |\langle A(u+tv)-A(u),w\rangle| \tag{Linearity of the duality pairing}\\ 209 | &\leq \|A(u+tv)-A(u)\|_{V'} \|w\|_V. \tag{Cauchy-Schwarz} 210 | \end{tightalign*} 211 | Thus, as $\|(u+tv)-u\|_V=\|tv\|\leq |t|\|u\|_V$, then $u+tv\overset{t\to 0}{\to} u$ in norm, and since $A$ is continuous, we get $A(u+tv)\overset{t\to 0}{\to} A(u)$ in $V'$. Thus, taking limit as $t\to 0$ we immediately conclude that $\varphi$ is continuous at $t=0$. 212 | \end{proof} 213 | \end{theorem} 214 | Now that we have defined hemicontinuous operators, we are ready to introduce a theorem that gives sufficient conditions that guarantee the surjectivity of a hemicontinuous monotone operator. 215 | \begin{theorem}[Minty-Browder]\label{thm:minty-browder} 216 | Let $V$ a real, separable and reflexive Banach space (e.g. $L^p$ for $p>1$), and $A:V\to V'$ a coercive and hemicontinuous monotone operator. Then, $A$ is surjective, i.e. given any $f\in V'$ there exists $u\in V$ such that $A(u)=f$. Moreover, if $A$ is strictly monotone, then $A$ is also injective, and thus there exists a unique solution for $A(u)=f$. 217 | \end{theorem} 218 | A common example of an operator that can be analyzed via the Minty-Browder theorem is the \emph{p-Laplacian} $-\Delta_p$, which for $p\geq 1$ is defined as the map 219 | \begin{tightalign*} 220 | -\Delta_p : W_0^{1,p}(\Omega)&\to W^{-1,q}(\Omega) = (W_0^{1,p}(\Omega))'\\ 221 | v&\mapsto -\Delta_p v = -\nabla\cdot(|\nabla v|^{p-2}\nabla v), 222 | \end{tightalign*} 223 | where $q$ is the conjugate exponent of $p$, such that $1/p + 1/q = 1$. Note that for arbitrary $u,v\in W_0^{1,p}(\Omega)$ the duality is given by 224 | \begin{equation*} 225 | \langle \Delta_p u, v\rangle = \langle \nabla\cdot (|\nabla u|^{p-2}\nabla u), v\rangle = -\langle |\nabla u|^{p-2}\nabla u, \nabla v\rangle = -\int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla v dx, 226 | \end{equation*} 227 | which is well-defined by Hölder's inequality: 228 | \begin{equation*} 229 | \left|\int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla v dx\right| \leq \|\nabla u\|^{p-1}_{W_0^{1,p}(\Omega)}\|\nabla v\|_{W_0^{1,p}(\Omega)}, 230 | \end{equation*} 231 | and noting that $w\mapsto \|\nabla w\|_{W_0^{1,p}(\Omega)}$ is a norm on $W_0^{1,p}(\Omega)$. We define now the functional 232 | \begin{tightalign*} 233 | \Psi:W_0^{1,p}(\Omega)&\to \R\\ 234 | u&\mapsto \Psi(u) \coloneqq \frac{1}{p}\int_\Omega |\nabla u|^p dx. 235 | \end{tightalign*} 236 | This functional is Gâteaux-differentiable, and we explicitly compute its the Gâteaux derivative for $u,v\in W_0^{1,p}(\Omega)$: 237 | \begin{tightalign*} 238 | d\Psi(u)[v] &= \left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0} \Psi(u+\varepsilon v)\\ 239 | &= \left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0} \frac{1}{p}\int_\Omega \underbrace{|\nabla (u+\varepsilon v)|^p}_{|\nabla(u+\varepsilon v)\cdot \nabla(u+\varepsilon v)|^{p/2}} dx\\ 240 | &= \left.\frac{1}{p}\int_\Omega \frac{p}{2} |\nabla (u+\varepsilon v)\cdot\nabla(u+\varepsilon v)|^{p/2-1}\cdot 2(\nabla u\cdot\nabla v) dx\right|_{\varepsilon = 0}\\ 241 | &= \int_\Omega |\nabla u\cdot\nabla u|^{\frac{p-2}{2}}\nabla u\cdot\nabla v dx\\ 242 | &= \int_\Omega |\nabla u|^{p-2}(\nabla u\cdot\nabla v) dx\\ 243 | &= \langle -\Delta_p u, v\rangle, 244 | \end{tightalign*} 245 | and thus $d\Psi = -\Delta_p$. This operator is hemicontinuous: take $t\in \R$ and $u,v,w\in W_0^{1,p}(\Omega)$, and define $\varphi(t) = \langle A(u+tv),w\rangle$. With this, we first compute 246 | $$ \varphi(t) - \varphi(0) = \int_\Omega \left(|\nabla (u+tv)|^{p-2}\nabla (u+tv) - |\nabla u|^{p-2}\nabla u\right)\cdot \nabla w dx \coloneqq \int_\Omega \Phi(t)\,dx, $$ 247 | and note that from the definitions of $u,v,w$ that \texttt{rhs} is bounded. This allows us to conclude from the Dominated Convergence theorem that 248 | $$ \lim_{t\to 0}\varphi(t) - \varphi(0) = \int_\Omega \lim_{t\to 0} \Phi(t)\,dx. $$ 249 | The limit within the integral is now trivial, from which we conclude the hemicontinuity of $-\Delta_p$. Moreover, since $\Psi$ is strictly convex, we obtain that for all $u\neq v \in W_0^{1,p}(\Omega)$, it holds that 250 | \begin{equation*} 251 | \langle \Delta_p u - \Delta_p v, u-v\rangle < 0 \implies \langle -\Delta_p u - (-\Delta_p v), u-v\rangle > 0, 252 | \end{equation*} 253 | which implies that $-\Delta_p$ is strictly monotone. To prove coercivity, we take a nonzero $u\in W_0^{1,p}(\Omega)$ and get 254 | \begin{tightalign*} 255 | \frac{\langle A(u),u\rangle}{\|u\|_{W_0^{1,p}(\Omega)}} &= \frac{1}{\|u\|_{W_0^{1,p}(\Omega)}}\int_\Omega \underbrace{|\nabla u|^{p-2}(\nabla u\cdot\nabla u)}_{|\nabla u|^p} dx\\ 256 | &= \frac{1}{\|u\|_{W_0^{1,p}(\Omega)}}\|u\|_{W_0^{1,p}(\Omega)}^p\\ 257 | &= \|u\|_{W_0^{1,p}(\Omega)}^{p-1}\xlongrightarrow{\|u\|_{W_0^{1,p}(\Omega)}\to\infty}\infty, 258 | \end{tightalign*} 259 | since $p>1$, which proves that $-\Delta_p$ is coercive. Since $W_0^{1,p}(\Omega)$ is separable and reflexive for $p>1$, the conditions of the Minty-Browder theorem~\ref{thm:minty-browder} are satisfied, and thus we conclude that $-\Delta_p$ is bijective, i.e. for any $f\in W^{-1,q}(\Omega)$ there exists a unique $u\in W_0^{1,p}(\Omega)$ such that $-\Delta_p u = f$. 260 | -------------------------------------------------------------------------------- /main.bib: -------------------------------------------------------------------------------- 1 | @article{AShortHistoryNieder2019, 2 | author = {Niederer, SA and Campbell, KS and Campbell, SG}, 3 | doi = {10.1016/j.yjmcc.2018.11.015}, 4 | journal = {Journal of Molecular and Cellular Cardiology}, 5 | language = {en}, 6 | month = {2}, 7 | pages = {11--19}, 8 | publisher = {Elsevier BV}, 9 | title = {A short history of the development of mathematical models of cardiac mechanics}, 10 | url = {http://dx.doi.org/10.1016/j.yjmcc.2018.11.015}, 11 | volume = {127}, 12 | year = {2019}, 13 | } 14 | 15 | @book{tortora2018principles, 16 | title={Principles of anatomy and physiology}, 17 | author={Tortora, GJ and Derrickson, BH}, 18 | year={2018}, 19 | publisher={John Wiley \& Sons} 20 | } 21 | 22 | @article{EffectOfTissuRobert1982, 23 | author = {Roberts, DE and Scher, AM}, 24 | doi = {10.1161/01.res.50.3.342}, 25 | issue = {3}, 26 | journal = {Circulation Research}, 27 | language = {en}, 28 | month = {3}, 29 | pages = {342--351}, 30 | publisher = {Ovid Technologies (Wolters Kluwer Health)}, 31 | title = {Effect of tissue anisotropy on extracellular potential fields in canine myocardium in situ.}, 32 | url = {http://dx.doi.org/10.1161/01.res.50.3.342}, 33 | volume = {50}, 34 | year = {1982}, 35 | } 36 | 37 | @inproceedings{HeartMuscleFiZhukovNone, 38 | author = {Zhukov, L and Barr, AH}, 39 | booktitle = {IEEE Visualization 2003}, 40 | doi = {10.1109/visual.2003.1250425}, 41 | journal = {IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control}, 42 | publisher = {IEEE}, 43 | title = {Heart-muscle fiber reconstruction from diffusion tensor {MRI}}, 44 | url = {http://dx.doi.org/10.1109/visual.2003.1250425}, 45 | venue = {Seattle, WA, USA}, 46 | year = {None}, 47 | } 48 | 49 | @article{FiberOrientatiStreet1969, 50 | author = {Streeter, DD and Spotnitz, HM and Patel, DP and Ross, J and Sonnenblick, EH}, 51 | doi = {10.1161/01.res.24.3.339}, 52 | issue = {3}, 53 | journal = {Circulation Research}, 54 | language = {en}, 55 | month = {3}, 56 | pages = {339--347}, 57 | publisher = {Ovid Technologies (Wolters Kluwer Health)}, 58 | title = {Fiber Orientation in the Canine Left Ventricle during Diastole and Systole}, 59 | url = {http://dx.doi.org/10.1161/01.res.24.3.339}, 60 | volume = {24}, 61 | year = {1969}, 62 | } 63 | 64 | @article{ANovelRuleBaBayer2012, 65 | author = {Bayer, JD and Blake, RC and Plank, G and Trayanova, NA}, 66 | doi = {10.1007/s10439-012-0593-5}, 67 | issue = {10}, 68 | journal = {Annals of Biomedical Engineering}, 69 | language = {en}, 70 | month = {10}, 71 | pages = {2243--2254}, 72 | publisher = {Springer Science and Business Media LLC}, 73 | title = {A Novel Rule-Based Algorithm for Assigning Myocardial Fiber Orientation to Computational Heart Models}, 74 | url = {http://dx.doi.org/10.1007/s10439-012-0593-5}, 75 | volume = {40}, 76 | year = {2012}, 77 | } 78 | 79 | @inbook{ModelingAtrialKruege2011, 80 | author = {Krueger, MW and Schmidt, V and Tob\'{o}n, C and Weber, FM and Lorenz, C and Keller, DUJ and Barschdorf, H and Burdumy, M and Neher, P and Plank, G and Rhode, K and Seemann, G and Sanchez-Quintana, D and Saiz, J and Razavi, R and D\"{o}ssel, O}, 81 | doi = {10.1007/978-3-642-21028-0\_28}, 82 | isbn = {['9783642210273', '9783642210280']}, 83 | journal = {Functional Imaging and Modeling of the Heart}, 84 | pages = {223--232}, 85 | publisher = {Springer Berlin Heidelberg}, 86 | title = {Modeling Atrial Fiber Orientation in Patient-Specific Geometries: A Semi-automatic Rule-Based Approach}, 87 | url = {http://dx.doi.org/10.1007/978-3-642-21028-0\_28}, 88 | year = {2011}, 89 | } 90 | 91 | @article{ModelingCardiaPiersa2021, 92 | author = {Piersanti, R and Africa, PC and Fedele, M and Vergara, C and Ded\`{e}, L and Corno, AF and Quarteroni, AM}, 93 | doi = {10.1016/j.cma.2020.113468}, 94 | journal = {Computer Methods in Applied Mechanics and Engineering}, 95 | language = {en}, 96 | month = {1}, 97 | pages = {113468}, 98 | publisher = {Elsevier BV}, 99 | title = {Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations}, 100 | url = {http://dx.doi.org/10.1016/j.cma.2020.113468}, 101 | volume = {373}, 102 | year = {2021}, 103 | } 104 | 105 | @article{AnAutomatePipZheng2021, 106 | author = {Zheng, T and Azzolin, L and S\'{a}nchez, J and D\"{o}ssel, O and Loewe, A}, 107 | doi = {10.1515/cdbme-2021-2035}, 108 | issue = {2}, 109 | journal = {Current Directions in Biomedical Engineering}, 110 | language = {en}, 111 | month = {10}, 112 | pages = {136--139}, 113 | publisher = {Walter de Gruyter GmbH}, 114 | title = {An automate pipeline for generating fiber orientation and region annotation in patient specific atrial models}, 115 | url = {http://dx.doi.org/10.1515/cdbme-2021-2035}, 116 | volume = {7}, 117 | year = {2021}, 118 | } 119 | 120 | @article{ATechniqueForRoney2019, 121 | author = {Roney, CH and Whitaker, J and Sim, I and O'Neill, L and Mukherjee, RK and Razeghi, O and Vigmond, EJ and Wright, M and O'Neill, MD and Williams, SE and Niederer, SA}, 122 | doi = {10.1016/j.compbiomed.2018.10.019}, 123 | journal = {Computers in Biology and Medicine}, 124 | language = {en}, 125 | month = {1}, 126 | pages = {278--290}, 127 | publisher = {Elsevier BV}, 128 | title = {A technique for measuring anisotropy in atrial conduction to estimate conduction velocity and atrial fibre direction}, 129 | url = {http://dx.doi.org/10.1016/j.compbiomed.2018.10.019}, 130 | volume = {104}, 131 | year = {2019}, 132 | } 133 | 134 | @article{PhysicsInformeRuizH2022, 135 | author = {Ruiz Herrera, C and Grandits, T and Plank, G and Perdikaris, P and Sahli Costabal, F and Pezzuto, S}, 136 | doi = {10.1007/s00366-022-01709-3}, 137 | issue = {5}, 138 | journal = {Engineering with Computers}, 139 | language = {en}, 140 | month = {10}, 141 | pages = {3957--3973}, 142 | publisher = {Springer Science and Business Media LLC}, 143 | title = {Physics-informed neural networks to learn cardiac fiber orientation from multiple electroanatomical maps}, 144 | url = {http://dx.doi.org/10.1007/s00366-022-01709-3}, 145 | volume = {38}, 146 | year = {2022}, 147 | } 148 | 149 | @article{ThermodynamicalRossi2014, 150 | author = {Rossi, S and Lassila, T and Ruiz-Baier, R and Sequeira, A and Quarteroni, AM}, 151 | doi = {10.1016/j.euromechsol.2013.10.009}, 152 | journal = {European Journal of Mechanics - A/Solids}, 153 | language = {en}, 154 | month = {11}, 155 | pages = {129--142}, 156 | publisher = {Elsevier BV}, 157 | title = {Thermodynamically consistent orthotropic activation model capturing ventricular systolic wall thickening in cardiac electromechanics}, 158 | url = {http://dx.doi.org/10.1016/j.euromechsol.2013.10.009}, 159 | volume = {48}, 160 | year = {2014}, 161 | } 162 | 163 | @article{GeneratingFibrWong2014, 164 | author = {Wong, J and Kuhl, E}, 165 | doi = {10.1080/10255842.2012.739167}, 166 | issue = {11}, 167 | journal = {Computer Methods in Biomechanics and Biomedical Engineering}, 168 | language = {en}, 169 | month = {8}, 170 | pages = {1217--1226}, 171 | publisher = {Informa UK Limited}, 172 | title = {Generating fibre orientation maps in human heart models using Poisson interpolation}, 173 | url = {http://dx.doi.org/10.1080/10255842.2012.739167}, 174 | volume = {17}, 175 | year = {2014}, 176 | } 177 | 178 | @article{ARuleBasedMeDoste2019, 179 | author = {Doste, R and Soto-Iglesias, D and Bernardino, G and Alcaine, A and Sebastian, R and Giffard-Roisin, S and Sermesant, M and Berruezo, A and Sanchez-Quintana, D and Camara, O}, 180 | doi = {10.1002/cnm.3185}, 181 | issue = {4}, 182 | journal = {International Journal for Numerical Methods in Biomedical Engineering}, 183 | language = {en}, 184 | month = {4}, 185 | pages = {e3185}, 186 | publisher = {Wiley}, 187 | title = {A rule-based method to model myocardial fiber orientation in cardiac biventricular geometries with outflow tracts}, 188 | url = {http://dx.doi.org/10.1002/cnm.3185}, 189 | volume = {35}, 190 | year = {2019}, 191 | } 192 | 193 | @inproceedings{AnimatingRotatShoema1985, 194 | author = {Shoemake, K}, 195 | booktitle = {the 12th annual conference}, 196 | doi = {10.1145/325334.325242}, 197 | journal = {Proceedings of the 12th annual conference on Computer graphics and interactive techniques - SIGGRAPH '85}, 198 | publisher = {ACM Press}, 199 | title = {Animating rotation with quaternion curves}, 200 | url = {http://dx.doi.org/10.1145/325334.325242}, 201 | venue = {Not Known}, 202 | year = {1985}, 203 | } 204 | 205 | @inbook{LiquidCrystalsBall2017, 206 | author = {Ball, JM}, 207 | doi = {10.1007/978-3-319-67600-5\_1}, 208 | isbn = {['9783319675992', '9783319676005']}, 209 | journal = {Mathematical Thermodynamics of Complex Fluids}, 210 | month = {9}, 211 | pages = {1--46}, 212 | publisher = {Springer International Publishing}, 213 | title = {Liquid Crystals and Their Defects}, 214 | url = {http://dx.doi.org/10.1007/978-3-319-67600-5\_1}, 215 | year = {2017}, 216 | } 217 | 218 | @article{TheNematicChiAuriau2022, 219 | author = {Auriau, J and Usson, Y and Jouk, P-S}, 220 | doi = {10.3390/jcdd9110371}, 221 | issue = {11}, 222 | journal = {Journal of Cardiovascular Development and Disease}, 223 | language = {en}, 224 | month = {10}, 225 | pages = {371}, 226 | publisher = {MDPI AG}, 227 | title = {The Nematic Chiral Liquid Crystal Structure of the Cardiac Myoarchitecture: Disclinations and Topological Singularities}, 228 | url = {http://dx.doi.org/10.3390/jcdd9110371}, 229 | volume = {9}, 230 | year = {2022}, 231 | } 232 | 233 | @article{LiquidCrystalsHirst2017, 234 | author = {Hirst, LS and Charras, G}, 235 | doi = {10.1038/544164a}, 236 | issue = {7649}, 237 | journal = {Nature}, 238 | language = {en}, 239 | month = {4}, 240 | pages = {164--165}, 241 | publisher = {Springer Science and Business Media LLC}, 242 | title = {Liquid crystals in living tissue}, 243 | url = {http://dx.doi.org/10.1038/544164a}, 244 | volume = {544}, 245 | year = {2017}, 246 | } 247 | 248 | @article{NonlinearTheorLinF1989, 249 | author = {Lin, F-H}, 250 | doi = {10.1002/cpa.3160420605}, 251 | issue = {6}, 252 | journal = {Communications on Pure and Applied Mathematics}, 253 | language = {en}, 254 | month = {9}, 255 | pages = {789--814}, 256 | publisher = {Wiley}, 257 | title = {Nonlinear theory of defects in nematic liquid crystals; Phase transition and flow phenomena}, 258 | url = {http://dx.doi.org/10.1002/cpa.3160420605}, 259 | volume = {42}, 260 | year = {1989}, 261 | } 262 | 263 | @article{ASaddlePointHuQi2009, 264 | author = {Hu, Q and Tai, X-C and Winther, R}, 265 | doi = {10.1137/060675575}, 266 | issue = {2}, 267 | journal = {SIAM Journal on Numerical Analysis}, 268 | language = {en}, 269 | month = {1}, 270 | pages = {1500--1523}, 271 | publisher = {Society for Industrial \& Applied Mathematics (SIAM)}, 272 | title = {A Saddle Point Approach to the Computation of Harmonic Maps}, 273 | url = {http://dx.doi.org/10.1137/060675575}, 274 | volume = {47}, 275 | year = {2009}, 276 | } 277 | 278 | @article{BlockPreconditBeik2018, 279 | author = {Beik, FPA and Benzi, M}, 280 | doi = {10.1007/s10092-018-0271-6}, 281 | issue = {3}, 282 | journal = {Calcolo}, 283 | language = {en}, 284 | month = {9}, 285 | publisher = {Springer Science and Business Media LLC}, 286 | title = {Block preconditioners for saddle point systems arising from liquid crystal directors modeling}, 287 | url = {http://dx.doi.org/10.1007/s10092-018-0271-6}, 288 | volume = {55}, 289 | year = {2018}, 290 | } 291 | 292 | @article{ConstrainedOptAdler2016, 293 | author = {Adler, JH and Emerson, DB and MacLachlan, SP and Manteuffel, TA}, 294 | doi = {10.1137/141001846}, 295 | issue = {1}, 296 | journal = {SIAM Journal on Scientific Computing}, 297 | language = {en}, 298 | month = {1}, 299 | pages = {B50--B76}, 300 | publisher = {Society for Industrial \& Applied Mathematics (SIAM)}, 301 | title = {Constrained Optimization for Liquid Crystal Equilibria}, 302 | url = {http://dx.doi.org/10.1137/141001846}, 303 | volume = {38}, 304 | year = {2016}, 305 | } 306 | 307 | @article{AugmentedLagraXiaJ2021, 308 | author = {Xia, J and Farrell, PE and Wechsung, F}, 309 | doi = {10.1007/s10543-020-00838-9}, 310 | issue = {2}, 311 | journal = {BIT Numerical Mathematics}, 312 | language = {en}, 313 | month = {6}, 314 | pages = {607--644}, 315 | publisher = {Springer Science and Business Media LLC}, 316 | title = {Augmented Lagrangian preconditioners for the {O}seen\textendash{}{F}rank model of nematic and cholesteric liquid crystals}, 317 | url = {http://dx.doi.org/10.1007/s10543-020-00838-9}, 318 | volume = {61}, 319 | year = {2021}, 320 | } 321 | 322 | @article{stenberg1995some, 323 | title={On some techniques for approximating boundary conditions in the finite element method}, 324 | author={Stenberg, R}, 325 | journal={Journal of Computational and applied Mathematics}, 326 | volume={63}, 327 | number={1-3}, 328 | pages={139--148}, 329 | year={1995}, 330 | publisher={Elsevier} 331 | } 332 | 333 | @article{ExistenceAndPHardt1986, 334 | author = {Hardt, R and Kinderlehrer, D and Lin, F-H}, 335 | doi = {10.1007/bf01238933}, 336 | issue = {4}, 337 | journal = {Communications in Mathematical Physics}, 338 | language = {en}, 339 | month = {12}, 340 | pages = {547--570}, 341 | publisher = {Springer Science and Business Media LLC}, 342 | title = {Existence and partial regularity of static liquid crystal configurations}, 343 | url = {http://dx.doi.org/10.1007/bf01238933}, 344 | volume = {105}, 345 | year = {1986}, 346 | } 347 | 348 | @book{dacorogna2014introduction, 349 | title={Introduction to the Calculus of Variations}, 350 | author={Dacorogna, B}, 351 | year={2014}, 352 | publisher={World Scientific Publishing Company} 353 | } 354 | 355 | @article{wright1999numerical, 356 | title={Numerical optimization}, 357 | author={Wright, S and Nocedal, J}, 358 | journal={Springer Science}, 359 | volume={35}, 360 | number={67-68}, 361 | pages={7}, 362 | year={1999} 363 | } 364 | 365 | @book{boffi2013mixed, 366 | title={Mixed finite element methods and applications}, 367 | author={Boffi, D and Brezzi, F and Fortin, M}, 368 | volume={44}, 369 | year={2013}, 370 | publisher={Springer} 371 | } 372 | 373 | @article{rathgeber2016firedrake, 374 | title={Firedrake: automating the finite element method by composing abstractions}, 375 | author={Rathgeber, F and Ham, DA and Mitchell, L and Lange, M and Luporini, F and McRae, ATT and Bercea, G-T and Markall, GR and Kelly, PHJ}, 376 | journal={ACM Transactions on Mathematical Software (TOMS)}, 377 | volume={43}, 378 | number={3}, 379 | pages={1--27}, 380 | year={2016}, 381 | publisher={ACM New York, NY, USA} 382 | } 383 | 384 | @inproceedings{falgout2002hypre, 385 | title={hypre: A library of high performance preconditioners}, 386 | author={Falgout, RD and Yang, UM}, 387 | booktitle={International Conference on Computational Science}, 388 | pages={632--641}, 389 | year={2002}, 390 | organization={Springer} 391 | } 392 | 393 | @article{MultigridMethoSchobe1999, 394 | author = {Sch\"{o}berl, J}, 395 | doi = {10.1007/s002110050465}, 396 | issue = {1}, 397 | journal = {Numerische Mathematik}, 398 | month = {11}, 399 | pages = {97--119}, 400 | publisher = {Springer Science and Business Media LLC}, 401 | title = {Multigrid methods for a parameter dependent problem in primal variables}, 402 | url = {http://dx.doi.org/10.1007/s002110050465}, 403 | volume = {84}, 404 | year = {1999}, 405 | } 406 | 407 | @article{barnafi2022analysis, 408 | title={Analysis and numerical validation of robust parallel nonlinear solvers for implicit time discretizations of the Bidomain equations}, 409 | author={Barnafi, NA and Huynh, NMM and Pavarino, LF and Scacchi, S}, 410 | journal={arXiv preprint arXiv:2209.05193}, 411 | year={2022} 412 | } 413 | 414 | @article{ParallelInexacBarnaf2022, 415 | author = {Barnafi, NA and Pavarino, LF and Scacchi, S}, 416 | doi = {10.1016/j.cma.2022.115557}, 417 | journal = {Computer Methods in Applied Mechanics and Engineering}, 418 | language = {en}, 419 | month = {10}, 420 | pages = {115557}, 421 | publisher = {Elsevier BV}, 422 | title = {Parallel inexact Newton\textendash{}Krylov and quasi-Newton solvers for nonlinear elasticity}, 423 | url = {http://dx.doi.org/10.1016/j.cma.2022.115557}, 424 | volume = {400}, 425 | year = {2022}, 426 | } 427 | 428 | @article{BiologicalTissSawT2018, 429 | author = {Saw, TB and Xi, W and Ladoux, B and Lim, CT}, 430 | doi = {10.1002/adma.201802579}, 431 | issue = {47}, 432 | journal = {Advanced Materials}, 433 | language = {en}, 434 | month = {11}, 435 | pages = {1802579}, 436 | publisher = {Wiley}, 437 | title = {Biological Tissues as Active Nematic Liquid Crystals}, 438 | url = {http://dx.doi.org/10.1002/adma.201802579}, 439 | volume = {30}, 440 | year = {2018}, 441 | } 442 | 443 | @article{LandauDeGenneMajumd2010, 444 | author = {Majumdar, A and Zarnescu, A}, 445 | doi = {10.1007/s00205-009-0249-2}, 446 | issue = {1}, 447 | journal = {Archive for Rational Mechanics and Analysis}, 448 | language = {en}, 449 | month = {4}, 450 | pages = {227--280}, 451 | publisher = {Springer Science and Business Media LLC}, 452 | title = {Landau\textendash{}{D}e {G}ennes Theory of Nematic Liquid Crystals: the {O}seen\textendash{}{F}rank Limit and Beyond}, 453 | url = {http://dx.doi.org/10.1007/s00205-009-0249-2}, 454 | volume = {196}, 455 | year = {2010}, 456 | } 457 | 458 | @article{OrientationOfTang2017, 459 | author = {Tang, X and Selinger, JV}, 460 | doi = {10.1039/c7sm01195d}, 461 | issue = {32}, 462 | journal = {Soft Matter}, 463 | language = {en}, 464 | pages = {5481--5490}, 465 | publisher = {Royal Society of Chemistry (RSC)}, 466 | title = {Orientation of topological defects in 2{D} nematic liquid crystals}, 467 | url = {http://dx.doi.org/10.1039/c7sm01195d}, 468 | volume = {13}, 469 | year = {2017}, 470 | } 471 | 472 | @article{Schoen1982regularity, 473 | title={A regularity theory for harmonic maps}, 474 | author={Schoen, R and Uhlenbeck, K}, 475 | journal={Journal of Differential Geometry}, 476 | volume={17}, 477 | number={2}, 478 | pages={307--335}, 479 | year={1982}, 480 | publisher={Lehigh University} 481 | } 482 | 483 | @inproceedings{Hardt1988stable, 484 | title={Stable defects of minimizers of constrained variational principles}, 485 | author={Hardt, R and Kinderlehrer, D and Lin, F-H}, 486 | booktitle={Annales de l'Institut Henri Poincar{\'e} C, Analyse non lin{\'e}aire}, 487 | volume={5}, 488 | number={4}, 489 | pages={297--322}, 490 | year={1988}, 491 | organization={Elsevier} 492 | } 493 | 494 | @book{Marsden2003vector, 495 | title={Vector calculus}, 496 | author={Marsden, JE and Tromba, A}, 497 | year={2003}, 498 | publisher={Macmillan} 499 | } 500 | 501 | @inproceedings{Pyop2AHighLRathge2012, 502 | author = {Rathgeber, F and Markall, GR and Mitchell, L and Loriant, N and Ham, DA and Bertolli, C and Kelly, PHJ}, 503 | booktitle = {2012 SC Companion: High Performance Computing, Networking, Storage and Analysis (SCC)}, 504 | doi = {10.1109/sc.companion.2012.134}, 505 | journal = {2012 SC Companion: High Performance Computing, Networking Storage and Analysis}, 506 | month = {11}, 507 | publisher = {IEEE}, 508 | title = {PyOP2: A High-Level Framework for Performance-Portable Simulations on Unstructured Meshes}, 509 | url = {http://dx.doi.org/10.1109/sc.companion.2012.134}, 510 | venue = {Salt Lake City, UT}, 511 | year = {2012}, 512 | } 513 | 514 | @article{lawson1979basic, 515 | title={Basic linear algebra subprograms for Fortran usage}, 516 | author={Lawson, CL and Hanson, RJ and Kincaid, DR and Krogh, FT}, 517 | journal={ACM Transactions on Mathematical Software (TOMS)}, 518 | volume={5}, 519 | number={3}, 520 | pages={308--323}, 521 | year={1979}, 522 | publisher={ACM New York, NY, USA} 523 | } 524 | 525 | 526 | @article{VerificationOfLand2015, 527 | author = {Land, S and Gurev, V and Arens, S and Augustin, CM and Baron, L and Blake, R and Bradley, C and Castro, S and Crozier, A and Favino, M and Fastl, TE and Fritz, T and Gao, H and Gizzi, A and Griffith, BE and Hurtado, DE and Krause, R and Luo, X and Nash, MP and Pezzuto, S and Plank, G and Rossi, S and Ruprecht, D and Seemann, G and Smith, NP and Sundnes, J and Rice, JJ and Trayanova, N and Wang, D and Jenny Wang, Z and Niederer, SA}, 528 | doi = {10.1098/rspa.2015.0641}, 529 | issue = {2184}, 530 | journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, 531 | language = {en}, 532 | month = {12}, 533 | pages = {20150641}, 534 | publisher = {The Royal Society}, 535 | title = {Verification of cardiac mechanics software: benchmark problems and solutions for testing active and passive material behaviour}, 536 | url = {http://dx.doi.org/10.1098/rspa.2015.0641}, 537 | volume = {471}, 538 | year = {2015}, 539 | } 540 | 541 | 542 | @article{VerificationOfNieder2011, 543 | author = {Niederer, SA and Kerfoot, E and Benson, AP and Bernabeu, MO and Bernus, O and Bradley, C and Cherry, EM and Clayton, R and Fenton, FH and Garny, A and Heidenreich, E and Land, S and Maleckar, M and Pathmanathan, P and Plank, G and Rodr\'{\i}guez, JF and Roy, I and Sachse, FB and Seemann, G and Skavhaug, O and Smith, NP}, 544 | doi = {10.1098/rsta.2011.0139}, 545 | issue = {1954}, 546 | journal = {Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences}, 547 | language = {en}, 548 | month = {11}, 549 | pages = {4331--4351}, 550 | publisher = {The Royal Society}, 551 | title = {Verification of cardiac tissue electrophysiology simulators using an 552 | N 553 | -version benchmark}, 554 | url = {http://dx.doi.org/10.1098/rsta.2011.0139}, 555 | volume = {369}, 556 | year = {2011}, 557 | } 558 | 559 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 560 | 561 | @book{quarteroni2008numerical, 562 | title={Numerical approximation of partial differential equations}, 563 | author={Quarteroni, A and Valli, A}, 564 | volume={23}, 565 | year={2008}, 566 | publisher={Springer Science \& Business Media} 567 | } 568 | 569 | @article{FiniteElementVerfr1986, 570 | author = {Verf\"u rth, R}, 571 | doi = {10.1007/bf01398380}, 572 | issue = {6}, 573 | journal = {Numerische Mathematik}, 574 | language = {en}, 575 | month = {11}, 576 | pages = {697--721}, 577 | publisher = {Springer Science and Business Media LLC}, 578 | title = {Finite element approximation on incompressible Navier-Stokes equations with slip boundary condition}, 579 | url = {http://dx.doi.org/10.1007/bf01398380}, 580 | volume = {50}, 581 | year = {1986}, 582 | } 583 | 584 | @article{NitschesMethoGjerde2021, 585 | author = {Gjerde, I and Scott, L}, 586 | doi = {10.1090/mcom/3682}, 587 | issue = {334}, 588 | journal = {Mathematics of Computation}, 589 | language = {en}, 590 | month = {11}, 591 | pages = {597--622}, 592 | publisher = {American Mathematical Society (AMS)}, 593 | title = {Nitsche's method for Navier\textendash{}Stokes equations with slip boundary conditions}, 594 | url = {http://dx.doi.org/10.1090/mcom/3682}, 595 | volume = {91}, 596 | year = {2021}, 597 | } 598 | 599 | @article{Chouly2024, 600 | title = {A Review on Some Discrete Variational Techniques for the Approximation of Essential Boundary Conditions}, 601 | ISSN = {2305-2228}, 602 | url = {http://dx.doi.org/10.1007/s10013-024-00702-1}, 603 | DOI = {10.1007/s10013-024-00702-1}, 604 | journal = {Vietnam Journal of Mathematics}, 605 | publisher = {Springer Science and Business Media LLC}, 606 | author = {Chouly, Franz}, 607 | year = {2024} 608 | } 609 | 610 | @book{ciarlet2013linear, 611 | title={Linear and nonlinear functional analysis with applications}, 612 | author={Ciarlet, PG}, 613 | year={2013}, 614 | publisher={SIAM} 615 | } 616 | 617 | @book{adams2003sobolev, 618 | title={Sobolev spaces}, 619 | author={Adams, RA and Fournier, JF}, 620 | year={2003}, 621 | publisher={Elsevier} 622 | } 623 | 624 | @article{gatica2014simple, 625 | title={A simple introduction to the mixed finite element method}, 626 | author={Gatica, GN}, 627 | journal={Theory and Applications. Springer Briefs in Mathematics. Springer, London}, 628 | year={2014}, 629 | publisher={Springer} 630 | } 631 | 632 | @book{monk2003finite, 633 | title={Finite element methods for Maxwell's equations}, 634 | author={Monk, P}, 635 | year={2003}, 636 | publisher={Oxford university press} 637 | } 638 | 639 | @book{brenner2008mathematical, 640 | title={The mathematical theory of finite element methods}, 641 | author={Brenner, SC and Scott, R}, 642 | year={2008}, 643 | publisher={Springer} 644 | } 645 | 646 | @book{evans2022partial, 647 | title={Partial differential equations}, 648 | author={Evans, LC}, 649 | volume={19}, 650 | year={2022}, 651 | publisher={American Mathematical Society} 652 | } 653 | 654 | @article{evans2009n, 655 | title={n-Widths, sup--infs, and optimality ratios for the k-version of the isogeometric finite element method}, 656 | author={Evans, JA and Bazilevs, Y and Babu{\v{s}}ka, I and Hughes, TJR}, 657 | journal={Computer Methods in Applied Mechanics and Engineering}, 658 | volume={198}, 659 | number={21-26}, 660 | pages={1726--1741}, 661 | year={2009}, 662 | publisher={Elsevier} 663 | } 664 | 665 | @book{ern2004theory, 666 | title={Theory and practice of finite elements}, 667 | author={Ern, A and Guermond, J-L}, 668 | volume={159}, 669 | year={2004}, 670 | publisher={Springer} 671 | } 672 | 673 | @article{warburton2003constants, 674 | title={On the constants in hp-finite element trace inverse inequalities}, 675 | author={Warburton, T and Hesthaven, JS}, 676 | journal={Computer methods in applied mechanics and engineering}, 677 | volume={192}, 678 | number={25}, 679 | pages={2765--2773}, 680 | year={2003}, 681 | publisher={Elsevier} 682 | } 683 | 684 | @book{sayas2019variational, 685 | title={Variational techniques for elliptic partial differential equations: Theoretical tools and advanced applications}, 686 | author={Sayas, FJ and Brown, TS and Hassell, ME}, 687 | year={2019}, 688 | publisher={CRC Press} 689 | } 690 | 691 | 692 | @article{moiola2021scattering, 693 | title={Scattering of time-harmonic acoustic waves: Helmholtz equation, boundary integral equations and BEM}, 694 | author={Moiola, A}, 695 | journal={Lecture notes for the “Advanced numerical methods for PDEs” class, University of Pavia, Department of Mathematics}, 696 | year={2021} 697 | } 698 | 699 | @book{BrezisFA, 700 | place={New York}, 701 | title={Functional analysis, Sobolev spaces and partial differential equations}, 702 | publisher={Springer}, 703 | author={Brezis, H.}, 704 | year={2011} 705 | } 706 | 707 | @book{ambrosetti1995primer, 708 | title={A primer of nonlinear analysis}, 709 | author={Ambrosetti, A and Prodi, G}, 710 | number={34}, 711 | year={1995}, 712 | publisher={Cambridge University Press} 713 | } 714 | 715 | @article{chen2024infSup, 716 | title = {Inf-sup Conditions for Operator Equations}, 717 | author = {Long Chen}, 718 | year = {2024}, 719 | journal = {Personal notes}, 720 | } 721 | 722 | @book{pata2019fixed, 723 | title={Fixed point theorems and applications}, 724 | author={Pata, V}, 725 | volume={116}, 726 | year={2019}, 727 | publisher={Springer} 728 | } 729 | 730 | @book{thomee2007galerkin, 731 | title={Galerkin finite element methods for parabolic problems}, 732 | author={Thom{\'e}e, V}, 733 | volume={25}, 734 | year={2007}, 735 | publisher={Springer Science \& Business Media} 736 | } 737 | 738 | 739 | @article{chenLFDM, 740 | title={Finite Difference Methods for Poisson Equation}, 741 | author={Chen, Long}, 742 | journal={Personal notes}, 743 | url={https://www.math.uci.edu/~chenlong/226/FDM.pdf}, 744 | year = 2020 745 | } 746 | 747 | @book{LeVeque2007, 748 | title = "Finite difference methods for ordinary and partial differential 749 | equations", 750 | author = "LeVeque, RJ", 751 | publisher = "Society for Industrial and Applied Mathematics", 752 | year = 2009 753 | } 754 | 755 | 756 | @book{Tao_2021, 757 | place={Providence, RI}, 758 | title={An introduction to measure theory}, 759 | publisher={American Mathematical Society}, 760 | author={Tao, Terence}, 761 | year={2021} 762 | } 763 | 764 | @book{Rudin_2013, 765 | place={New York, NY}, 766 | title={Real and complex analysis}, 767 | publisher={McGraw-Hill}, 768 | author={Rudin, Walter}, 769 | year={2013} 770 | } -------------------------------------------------------------------------------- /chapters/time-dependent.tex: -------------------------------------------------------------------------------- 1 | Our analysis so far has focused on stationary problems. We now turn our attention to the more general case of time-dependent problems, where the solution $u(t,x)$ evolves over a time interval $[0,T]$. The inclusion of a time derivative fundamentally alters the nature of the governing equations, leading to a rich classification of physical behaviors. 2 | 3 | This chapter introduces framework for analyzing such evolution problems. We begin by distinguishing between two primary classes of time-dependent systems based on their energy behavior: \emph{parabolic} systems, which are dissipative and model processes like heat diffusion, and \emph{hyperbolic} systems, which are conservative and describe wave propagation. 4 | 5 | To rigorously formulate these problems, we must extend our functional analysis toolkit to handle functions that map time to a function space. This leads naturally to the theory of \emph{Bochner spaces}, such as $L^2(0,T;V)$, which provide the proper setting for our analysis. With this machinery in place, we will establish the well-posedness of parabolic problems using the powerful \emph{Faedo-Galerkin method}. This constructive proof serves as the foundation for the numerical discretization strategy known as the \emph{method of lines}, where we first discretize in space to obtain a large system of ordinary differential equations, and then apply a subsequent time-stepping scheme to obtain a fully-discrete solution. This chapter follows the presentations from~\cite{thomee2007galerkin,quarteroni2008numerical}. 6 | 7 | \section{Parabolic and hyperbolic systems}\label{sec:parabolic-hyperbolic-systems} 8 | From now on, our time-dependent solutions are functions $V\ni u:[0,T]\times \Omega\to \R$. In this section, we consider $A:V\to V'$ an elliptic operator. 9 | 10 | \begin{definition}[Parabolic and hyperbolic generic operators]\label{def:parabolic-hyperbolic-operators} 11 | Let $\partial_t$ and $\partial_{tt}$ represent the first and second order time derivative operators, respectively. Then, we say that 12 | \begin{tightalign*} 13 | \partial_t + A &\quad \text{is parabolic,}\\ 14 | \partial_{tt} + A &\quad \text{is hyperbolic.} 15 | \end{tightalign*} 16 | \end{definition} 17 | The fundamental difference between parabolic and hyperbolic systems is the behaviour of their \emph{energy} $E(t)$. Let us formally derive this fact. 18 | \begin{enumerate} 19 | \item Parabolic systems: we can readily write the weak form of the parabolic equation $(\partial_t + A)u=0$ as 20 | \begin{equation} 21 | (\partial_t u, v) + (Au, v) = 0 \qquad \forall v\in V. 22 | \end{equation} 23 | We note that this weak formulation seems unbalanced as it is being tested only on the space variable. We will accept this for now, but see later on that it is indeed a good weak formulation for analyzing the problem using our knowledge of elliptic and Gårding operators. Since this is true for all $v\in V$, we can choose $v=u$, and we obtain 24 | \begin{equation*} 25 | (\partial_t u, u) + (Au,u) = 0. 26 | \end{equation*} 27 | Note that $\partial_t(u^2) = 2u\dot{u}$, and thus we can write $(\partial_t u, u) = \int_\Omega u\dot{u} = \frac{1}{2}\int_\Omega \partial_t (u^2)$, which leads to 28 | \begin{equation*} 29 | \frac{1}{2}\int_\Omega \partial_t (u^2) + \underbrace{\int Au \cdot u}_{ \coloneqq a(u,u)} = 0, 30 | \end{equation*} 31 | and integrating in time in $[0,t]$ with $t\leq T$ we get 32 | \begin{equation*} 33 | \frac{1}{2} \int_\Omega (u(t)^2-u(0)^2) + \int_0^T a(u,u)ds = 0, 34 | \end{equation*} 35 | where $u(0) = u(0,x)$ is a (fixed) initial condition. This implies that 36 | \begin{equation*} 37 | \frac{1}{2}\int_\Omega u(t)^2 = \frac{1}{2} \int_\Omega u(0)^2 - \int_0^T a(u,u)ds. 38 | \end{equation*} 39 | Defining the energy as $E(t) \coloneqq \int_\Omega u(t)^2$, we get 40 | \begin{equation} 41 | \frac{1}{2}E(t) = \frac{1}{2}E(0) - \int_0^T \underbrace{a(u,u)}_{\geq \alpha\|u\|^2>0} ds \implies \boxed{E(t) < E(0)}. 42 | \end{equation} 43 | We observe that energy decreases from the initial condition in a parabolic system. Thus, parabolic systems are called \emph{dissipative}. 44 | \item Hyperbolic systems: since we need a second-order time derivative, we note that $\partial_t(\dot{u}^2) = 2\dot{u}\ddot{u}$, and thus as before we can write 45 | \begin{equation} 46 | \int_\Omega \ddot{u}v + \int_\Omega A u\cdot v = 0 \qquad \forall v\in V. 47 | \end{equation} 48 | Setting $v=\dot{u}$, we obtain 49 | \begin{equation} 50 | \frac{1}{2}\int_\Omega \partial_t (\dot{u}^2) + \int_\Omega Au\cdot \dot{u} = 0 \qquad \forall v\in V. 51 | \end{equation} 52 | We now restrict ourselves to elliptic operators $A$ that can be written as $A=B^\top B$, such that $Au\cdot v = Bu\cdot Bv$. This way, we see that 53 | \begin{equation} 54 | \partial_t (Bu)^2 = 2Bu \cdot B\dot{u} = 2Au\cdot \dot{u}. 55 | \end{equation} 56 | As before, we integrate in time and get 57 | \begin{equation} 58 | \frac{1}{2} \int_\Omega \left[\dot{u}^2(t) + (Bu(t))^2\right] = \frac{1}{2} \int_\Omega \left[\dot{u}^2(0) + (Bu(0))^2\right]. 59 | \end{equation} 60 | Now, setting the energy as $E(t) \coloneqq \int_\Omega \left[u(t)^2 + (Bu(t))^2\right]$, we conclude that $E(t) = E(0)$. Thus, hyperbolic systems are \emph{conservative}. 61 | \end{enumerate} 62 | Our generic, parabolic initial boundary value problem is given by 63 | \begin{equation}\label{eq:parabolic-IBVP} 64 | \begin{aligned} 65 | \partial_t + Au &= f &&\quad \tin\Omega_T \coloneqq (0,T) \times \Omega\\ 66 | \hfill Bu &= g &&\quad \tin\Sigma_T \coloneqq (0,T) \times \partial\Omega\\ 67 | \hfill u(0,x)&= u_0(x) &&\quad \tin\Omega, 68 | \end{aligned} 69 | \end{equation} 70 | with $f,g:\Omega_T\to \R$ and $u_0:\Omega\to\R$. We can analyze how to deal with the time dependence of our system by introducing the \emph{Bochner integral}. 71 | 72 | \section{The Bochner integral}\label{sec:bochner} 73 | We seek to integrate functions $f:\R\to X$, where $X$ is a Banach space. To this end, we need to redefine our notion of simple functions. 74 | \begin{definition}[Bochner integral]\label{def:bochner-integral} 75 | We seek to integrate functions We extend the notion of simple functions introduced in~\ref{def:simple-functions} as 76 | \begin{equation} 77 | f^N(t) = \sum_{i=1}^N \lambda_i \phi_i(t), 78 | \end{equation} 79 | where $\lambda_i \in X$ $\forall i\in\{1,\dots,N\}$, and $\phi_i(t)$ are indicator functions. If $I\subset \R$, then the time integral yields 80 | \begin{equation} 81 | \int_I f^N(t) dt = \sum_{i=1}^N \lambda_i \int_I \phi_i(t) dt. 82 | \end{equation} 83 | Here, if we have absolute convergence of the corresponding series, i.e. 84 | \begin{equation} 85 | \sum_{i=0}^{\infty} \|\lambda_i\|_X \int_I \phi_i(t)dt <\infty, 86 | \end{equation} 87 | and if $f(t) = \sum_{i=1}^{\infty} \lambda_i \phi_i(t)$ for every $t$ where the series converges, then we say $f$ is \emph{Bochner integrable}, and 88 | \begin{equation} 89 | \int_0^T f(s)ds \coloneqq \sum_{i=1}^\infty \lambda_i \int_0^T \phi_i(s)ds. 90 | \end{equation} 91 | \end{definition} 92 | \begin{corollary} 93 | If $f$ is Bochner integrable, then $|f|$ is Lebesgue integrable. 94 | \end{corollary} 95 | 96 | We now define the \emph{Bochner spaces} we will be using for our analysis: 97 | \begin{tightalign} 98 | L^p(0,T; X) &\coloneqq \left\{v:(0,T)\to X: v \text{ is Bochner integrable}, \int_0^T \|v\|_X^p ds <\infty \right\}\\ 99 | H^1(0,T; X) &\coloneqq \left\{v:(0,T)\to X: v\in L^2(0,T;X), \partial_t v\in L^2(0,T;X) \right\}. 100 | \end{tightalign} 101 | In general, $\partial_t v$ should be interpreted as an element of $V'$, since the weak form we will be using is, given $f:\Omega_T\to \R$, 102 | \begin{equation} 103 | (\dot{u},v) + a(u,v) = \langle f, v\rangle \quad \forall v\in V, 104 | \end{equation} 105 | and thus $\dot{u}\in V'$ for the first term to exist. The natural definition that gives sense to this object is as follows: setting $X=L^2(\Omega)$ and $L^2(\Omega_T) \coloneqq L^2(0,T; L^2(\Omega))$, we denote 106 | \begin{equation} 107 | H^1(0,T;L^2(\Omega)) \coloneqq \left\{v:(0,T)\to X: v\in L^2(\Omega_T), \partial_t v\in L^2(\Omega_T) \right\}. 108 | \end{equation} 109 | This yields a norm equivalence: 110 | \begin{tightalign*} 111 | \|v\|_{L^2(\Omega_T)} &= \int_0^T \|v(s)\|_{0,\Omega}^2 ds\\ 112 | &= \int_0^T \left(\left(\int_\Omega v(s,x)^2 dx\right)^{1/2}\right)^2ds\\ 113 | &= \int_0^T \int_\Omega v^2 dxds\\ 114 | &= \int_{\Omega_T} v^2\\ 115 | &= \|v\|^2_{L^2(\Omega_T)}. 116 | \end{tightalign*} 117 | We now state an embedding theorem for Bochner spaces. 118 | \begin{theorem}\label{thm:embedding-bochner} 119 | Let $\Omega\subseteq \R^n$ be a Lipschitz domain, $s\geq 0$ and $r>1/2$. For any $\theta\in[0,1]$, the following embedding is continuous: 120 | \begin{equation} 121 | L^2(0,T;H^s(\Omega))\cap H^r(0,T;L^2(\Omega)) \longrightarrow H^{\theta r}(0,T;H^{(1-\theta)s}(\Omega))\cap C^0(0,T;H^{\sigma_0}(\Omega)), 122 | \end{equation} 123 | with $\sigma_0 = \frac{(2r-1)s}{2r}$. Furthermore, if $s>0$ and $|\Omega|<\infty$, then the following embedding is compact: 124 | \begin{equation} 125 | L^2(0,T;H^s(\Omega))\cap H^r(0,T;L^2(\Omega)) \longrightarrow H^{r_1}(0,T;H^{s_1}(\Omega))\cap C^0(0,T;H^{\sigma_1}(\Omega)), 126 | \end{equation} 127 | for any $s_1\geq 0$, $0\leq r_1 < r(1-s_1/s)$ and $0\leq \sigma_1 < \sigma$. 128 | \end{theorem} 129 | 130 | \section{Faedo-Galerkin and the method of lines}\label{sec:faedo-galerkin} 131 | In this section we seek to generalize the error analysis we performed for stationary PDEs via Galerkin schemes. We will limit ourselves to analyzing the parabolic initial boundary value problem~\eqref{eq:parabolic-IBVP} with an elliptic operator, and in a finite interval in time, assuming that the Dirichlet condition is $g=0$ for simplicity. 132 | 133 | In a continuous setting, for $f\in L^2(\Omega_T)$ and $u_0\in L^2(\Omega)$, we seek a solution $u\in L^2(0,T;V)\cap C([0,T];V)$ such that 134 | \begin{equation}\label{eq:weak-form-continuous-parabolic-IBVP} 135 | \begin{aligned} 136 | (\partial_t u(t), v) + a(u(v), v) &= \langle f(t), v\rangle\quad \forall v\in V,\\ 137 | u(0) &= u_0. 138 | \end{aligned} 139 | \end{equation} 140 | Note that we simplify the notation for the duality pairing $\langle f, v\rangle = \langle f, v\rangle_{V'\times V}$, since~\ref{eq:parabolic-IBVP} is understood distributionally. 141 | \paragraph{The semi-discrete problem} 142 | As with stationary problems, we can prove a continuous a priori estimate. To derive it, we will work on a finite-dimensional approximation and take weak limits at the end, which will require solving a finite-dimensional initial value problem (i.e. a system of ODEs), where the existence and uniqueness of solutions is justified by \emph{Carathéodory's existence theorem}\footnote{This theorem essentially relaxes the conditions that other stronger existence and/or uniqueness theorems require for the right hand side $\vec g$, as we now need continuity in $t$ and measurability in $\vec y$. In this sense, Carathéodory's existence theorem is a relaxation of the Peano existence theorem, which requires continuity in $t$ and $\vec y$, and this is itself a relaxation of the well-known Picard-Lindelöf existence and uniqueness theorem, which requires continuity in $t$ and Lipschitz continuity in $\vec y$.}. 143 | \begin{theorem}[Carathéodory's existence theorem]\label{thm:caratheodory-existence} 144 | Let $R = \{(t,y)\in \R\times \R^n: |t-t_0|\leq a, \|y-y_0\|\leq b\}$ be a rectangular domain. Consider the initial value problem 145 | \begin{equation}\label{eq:caratheodory-problem} 146 | \begin{aligned} 147 | \dot{\vec y} &= \vec g(t, \vec y) &&\tin R\\ 148 | \vec y(0) &= \vec y_0. 149 | \end{aligned} 150 | \end{equation} 151 | If the function $\vec g$ is such that 152 | \begin{enumerate} 153 | \item the map $\vec y \mapsto \vec g(t, \vec y)$ is continuous for each fixed $t$, 154 | \item the map $t\mapsto \vec g(t, \vec y)$ is measurable for each fixed $\vec y$, and 155 | \item $\vec g$ is dominated in $\vec L^1(0,T;\R^N)$, i.e. there exists a Lebesgue integrable function $m:[t_0-a,t_0+a]\to\R^+$ such that $\|f(t, \vec y)\|\leq m(t)$ for all $y\in\R^n$ with $\|y-y_0\|\leq b$, 156 | \end{enumerate} 157 | then a solution to~\ref{eq:caratheodory-problem} exists. Moreover, if there exists a function $k:[t_0-a,t_0+a]\to \R^+$ such that for all $(t,\vec y_1),(t,\vec y_2)\in\R$, it holds that 158 | \begin{equation*} 159 | \|\vec g(t, \vec y_1) - \vec g(t,\vec y_2)\| \leq k(t)\|\vec y_1-\vec y_2\|, 160 | \end{equation*} 161 | then the solution is unique. 162 | \end{theorem} 163 | With this theorem at hand, we are ready to prove the a priori estimate. 164 | \begin{theorem}[A priori estimate, time-dependent parabolic problem]\label{thm:a-priori-continuous-time-dependent} 165 | If $a$ is an elliptic form, then there exists a unique $u\in L^2(0,T;V)\cap C([0,T];V)$, such that $\partial_t u\in L^2(0,T;V)$ and 166 | \begin{equation} 167 | \|u(t)\|_0^2 + \alpha \int_0^T \|u(s)\|_V^2 ds \leq \|u_0\|_0^2 + \frac{1}{\alpha} \int_0^T \|f(s)\|^2 ds. 168 | \end{equation} 169 | \begin{proof} 170 | Let $V^N\subset V$ be a finite-dimensional subspace of $V$ with $\dim V^N = N$ and let $\{\phi_j\}_{j=1}^N$ be a fixed basis. We approximate the solution $u$ as 171 | \begin{equation*} 172 | u\approx u^N \coloneqq \sum_{j=1}^N u_j(t)\phi_j. 173 | \end{equation*} 174 | Then, the weak formulation~\eqref{eq:weak-form-continuous-parabolic-IBVP} results in the system of $N$ equations 175 | \begin{equation*}\label{eq:semi-discrete} 176 | (\partial_t u^N, \phi_i) + a(u^N, \phi_i) = \langle f(t), \phi_i\rangle \quad \forall i=1,\dots, N. 177 | \end{equation*} 178 | Similar to the stiffness matrix and force vector defined in~\eqref{eq:stiffness-force-fem}, we define 179 | \begin{equation*} 180 | M_{ij} \coloneqq (\phi_j, \phi_i),\quad A_{ij} \coloneqq a(\phi_j, \phi_i),\quad F_i \coloneqq \langle f(t), \phi_i\rangle, 181 | \end{equation*} 182 | where we now denote $\vec u = (u_1(t),\dots,u_N(t))$ the discrete coordinate vector of $u^N$. With this notation, we can rewrite~\eqref{eq:weak-form-continuous-parabolic-IBVP} as the discrete, linear system of ODEs on $\vec u$: 183 | \begin{equation}\label{eq:weak-parabolic-IBVP-discrete} 184 | \begin{aligned} 185 | \ten M \dot{\vec u} + \ten A \vec u &= \vec F\\ 186 | \vec u(0) &= \Pi_{V^N}(\vec u_0), 187 | \end{aligned} 188 | \end{equation} 189 | where $\Pi_{V^N}$ is the projection operator $\Pi_{V^N}:V\to V^N$. We can prove that all conditions in Carathéodory's existence theorem~\ref{thm:caratheodory-existence} are met, with $\vec y = \vec u$, $t_0=0$ and $\vec y_0=\Pi_{V^N}\vec u_0$, by taking 190 | \begin{equation*} 191 | \vec g(t, \vec u) \coloneqq \ten M^{-1}(\vec F(t)-\ten A\vec u), 192 | \end{equation*} 193 | which is well-defined since $\ten M$ is invertible due to the linear independence of the basis. Thus, there exists a unique solution $\dot{\vec u}$ to equation to~\eqref{eq:weak-parabolic-IBVP-discrete}. 194 | 195 | Now, since the discrete right hand side is $\vec F\in L^2(0,T;\R^N)$, then $\dot{\vec u} = \ten M^{-1}(\vec F(t)-\ten A\vec u)$ is in $L^2$, and thus $u^N\in \vec H^1(0,T;\R^N)$ and $u\in \vec H^1(0,T;V)$. By taking $\phi_i = u^N$ in~\eqref{eq:semi-discrete}, we obtain 196 | \begin{equation*} 197 | (\dot{u}^N, u^N) + a(u^N,u^N) = \langle f, u^N\rangle, 198 | \end{equation*} 199 | and using the identity $u\dot{u} = \frac{1}{2}\partial_t(u^2)$, and integrating in time between $0$ and $T$ we get 200 | \begin{equation*} 201 | \frac{1}{2}\left(\|u^N\|^2 - \|u_0^N\|^2\right) + \int_0^T a(u^N,u^N) ds = \int_0^T \langle f, u^N\rangle ds. 202 | \end{equation*} 203 | The ellipticity property allows us to bound $\int_0^T a(u^N,u^N)ds \geq \alpha\int_0^T \|u^N\|^2 ds$, and bounding the product $\langle f, u^N\rangle\leq \|f\|_{V'}\|u^N\|_{V^N}$, we get 204 | \begin{equation*} 205 | \frac{1}{2}\|u^N\|^2_{V^N} + \alpha \int_0^T \|u^N\|_V^2 ds \leq \frac{1}{2}\|u_0^N\|^2_{V^N} + \int_0^T \|f\|_{V'}\|u^N\|_{V^N} ds. 206 | \end{equation*} 207 | To bound the last term, we invoke Young's inequality~\eqref{thm:young-inequality} with $a=\|f\|_{V'}$, $b=\|u^N\|_{V^N}$ and $\varepsilon=\alpha$ (i.e. the ellipticity constant), where we get 208 | \begin{equation*} 209 | \|f\|_{V'}\|u^N\|_{V^N} \leq \frac{\|f\|_{V'}^2}{2\alpha} + \frac{\alpha \|u^N\|_{V^N}}{2}, 210 | \end{equation*} 211 | and substituting, factoring and multiplying by $2$, we get 212 | \begin{equation*} 213 | \|u^N\|^2_{V^N} + \alpha \int_0^T \|u^N\|_V^2 ds \leq \|u_0^N\|^2_{V^N} + \frac{1}{\alpha}\int_0^T \|f(s)\|_{V'} ds. 214 | \end{equation*} 215 | 216 | Note that $u^N$ is bounded in $L^\infty(0,T;L^2)$ and in $L^2(0,T;V)$, and by the weak$-^*$ convergence we can conclude that there exists a weak limit $u\in L^\infty(0,T;L^2)\cap L^2(0,T;V)$ for $u^N$ and its gradient $\nabla u^N$, such that 217 | \begin{tightalign*} 218 | \int_0^T (u^N(t),\varphi(t)) ds &\to \int_0^T (u(t), \varphi(t)) ds \qquad \forall \varphi\in L^1(0,T; L^2)\\ 219 | \int_0^T (\nabla u^N(t), \Phi(t))ds &\to \int_0^T (\nabla u(t), \Phi(t))ds \qquad \forall \Phi\in L^2(0,T; L^2). 220 | \end{tightalign*} 221 | 222 | We can take $\Psi\in C^1([0,T])$ such that $\Psi(T)= 0$, such that for all test functions $ v^N\in V^N$, the weak form 223 | \begin{equation*} 224 | (\dot{u}^N), v^N) + a(u^N, v^N) = \langle f, v^N\rangle 225 | \end{equation*} 226 | can be equivalently written without compromising the time dependence as 227 | \begin{equation*} 228 | (\dot{ u}^N), \Psi v^N) + a(u^N, \Psi v^N) = \langle f, \Psi v^N\rangle. 229 | \end{equation*} 230 | Integrating between $0$ and $T$, and integrating by parts, we get that for all $ v^N\in V^N$, 231 | \begin{tightalign*} 232 | \cancel{(u^N(T),\Psi(T) v^N)} - (u^N(0),\Psi(0) v^N) + \int_0^T a(u^N,\Psi v^N) ds &- \int_0^T (u^N, v^N)\Psi'(t)ds\\ 233 | &\quad = \int_0^T \langle f, \Psi v^N\rangle ds. 234 | \end{tightalign*} 235 | Here, note that $\Psi(t) v^N\in L^2(0,T;V)\cap L^1(0,T; L^2)$ because $\Psi(t) v^N\in C^1([0,T];V)$, and by the weak$-^*$ convergence, we have 236 | \begin{tightalign*} 237 | \int_0^T \langle f, \Psi v^N\rangle ds = -\int_0^T (u^N(0), \Psi(0)v^N)&ds - \int_0^T (u^N, v^N)\Psi'(t) ds + \int_0^T a(u^N, \Psi v^N) ds\\ 238 | &\big\downarrow \text{ weakly}\\ 239 | -\int_0^T ( u(0),\Psi(0) v^N)ds &-\int_0^T (u, v^N)\Psi'(t)ds + \int_0^T a( u, \Psi v^N)ds. 240 | \end{tightalign*} 241 | By density of $V^N$ in $V$, we have 242 | \begin{equation*} 243 | |a(u, v) - a(u, v^N)|\to 0, 244 | \end{equation*} 245 | which implies that for all $v \in V$, we get 246 | \begin{equation*} 247 | -( u(0),\Psi(0) v) - \int_0^T (u, v)\Psi' ds + \int_0^T a(u, \Psi v)ds = \int_0^T \langle f, \Psi v^N\rangle ds \qquad \forall v\in V,\forall \Psi. 248 | \end{equation*} 249 | Integrating by parts, we get 250 | \begin{equation*} 251 | \int_0^T (\dot{ u}, \Psi v) ds + \int_0^T a( u, \Psi v)ds = \int_0^T \langle f, \Psi \rangle ds \qquad \forall v\in V,\forall \Psi, 252 | \end{equation*} 253 | and thus since this result holds for all $\Psi\in C^1([0,T])$, it holds for distributions $\D((0,T])$. Carrying all terms to the left hand side, we get 254 | \begin{equation*} 255 | \int_0^T (\dot{u}, \Psi v) ds + \int_0^T a( u, \Psi v)ds -\int_0^T \langle f, \Psi v\rangle ds = 0,. 256 | \end{equation*} 257 | By the bilinearity of $a$ we note that $a(u, \Psi v) = \Psi(t)a(u, v)$, and by linearity of the inner product we can take $\Psi$ outside the inner products, to get 258 | \begin{equation*} 259 | \int_0^T \Psi\left[(\dot{u}, v) + a(u, v) - \langle f, v\rangle \right] ds = 0. 260 | \end{equation*} 261 | Borrowing the localization theorem~\ref{thm:localization} from the next chapter, since $\Psi$ is arbitrary, the above equality implies 262 | \begin{equation*} 263 | (\dot{u}, v) + a(u, v) - \langle f, v\rangle = 0 \qquad \forall v\in V. 264 | \end{equation*} 265 | 266 | To have $u(0) = u_0$, we can just take $\Psi(0)=1$, such that 267 | \begin{equation*} 268 | (u(0)-u_0^N, \Psi(0) v) = (u(0)-u_0^N, v) \to 0, 269 | \end{equation*} 270 | and the regularity of $\partial_t u\in L^2(V')$ can be derived from the continuity of $a$: note that 271 | \begin{tightalign*} 272 | (\partial_t u, v) &= \langle f, v\rangle -a(u,v)\\ 273 | &\leq \|f\|_{V'}\|v\|_V + C_{\text{cont}}^\alpha \|u\|_V\|v\|_V \tag{Cauchy-Schwarz, continuity of $a$} \\ 274 | &\leq C(u)\|v\|_V \tag{continuity of $F$}, 275 | \end{tightalign*} 276 | and taking supremum over $v$ we get $\|\partial_t u\|_{V'}\leq C(u)$, thus $\partial_t u$ is bounded and, in particular, in $L^2(V')$. 277 | \end{proof} 278 | \end{theorem} 279 | 280 | Now that we have an a priori estimate, we can analyze the convergence of a discrete method that involves a finite-dimensional discrete space $V_h$. In what follows, for any given bilinear form $a:V\times V\to \R$ we denote the \emph{Ritz projector}\footnote{Another common choice for the projector is the \emph{Scott-Zhang projector}.} as the operator $R_h:V\to V_h$ such that 281 | \begin{equation} 282 | a(R_h z, v_h) = a(z, v_h) \qquad \forall v_h \in V_h. 283 | \end{equation} 284 | In particular, the a priori bound yields 285 | \begin{equation} 286 | \|R_h z \| \leq \| z \|. 287 | \end{equation} 288 | 289 | We first prove the following convergence result for the semi-discrete problem, similar to Céa's estimate~\ref{eq:cea-estimate-fem}: 290 | \begin{theorem}\label{thm:convergence-semidiscrete-time} 291 | In problem~\eqref{eq:semi-discrete}, if both $u(t)$ and $\dot u(t)$ belong to $H^1(\Omega)$ for all fixed $t$ in $(0,T)$, then for $u_h\in V_h$, we have 292 | \begin{equation} 293 | \| u(t) - u_h(t) \|_V \leq \|u_0 - \Pi_h(u_0)\|_v + Ch^r\left(\|u_0\|_r+\int_0^T\|\dot u(s)\|_r\,ds\right). 294 | \end{equation} 295 | \begin{proof} 296 | The methodology consists in separating the error into \emph{projection} and \emph{consistency} errors\footnote{A common way for deriving this type of estimate is using the \emph{Gronwall inequality}.}: 297 | \begin{equation*} 298 | e_h \coloneqq u - u_h = \underbrace{u - \Pi_h u}_{\xi_h} + \underbrace{\Pi_h u - u_h}_{\eta_h} = \xi_h + \eta_h, 299 | \end{equation*} 300 | where $\xi_h$ is the projection error and $\eta_h$ is the consistency error. We now consider the error equation 301 | \begin{equation*} 302 | (\dot e_h, v_h) + a(e_h, v_h) = 0 \qquad \forall v_h\in V_h, 303 | \end{equation*} 304 | and we consider as the projector the Ritz projector, i.e. $\Pi_h=R_h$, which gives 305 | \begin{equation*} 306 | (\dot \xi_h + \dot \eta_h, v_h) + a(\eta_h, v_h) = 0 \qquad \forall v_h\in V_h. 307 | \end{equation*} 308 | Setting $v_h = \eta_h$, we obtain 309 | \begin{equation*} 310 | \frac 1 2 \partial_t(\|\eta_h\|^2_0)+a(\eta_h,\eta_h) = -(\dot \xi_h, v_h). 311 | \end{equation*} 312 | Note that 313 | \begin{equation*} 314 | \frac 1 2 \partial_t(\|\eta_h\|_0^2) = \|\eta_h\|_0 \partial_t(\|\eta_h\|_0), 315 | \end{equation*} 316 | and thus using the ellipticity of $a$ we obtain the following: 317 | \begin{equation*} 318 | \|\eta_h\|\partial_t(\|\eta_h\|_0) \leq \partial_t(\|\eta\|_0^2) + a(\eta_h, \eta_h) \leq \|\dot \eta_h\|_0 \|v_h\|_0, 319 | \end{equation*} 320 | which after dividing by $\|v_h\|_0$ and integrating between $0$ and $T$ yields 321 | \begin{equation*} 322 | \| \eta_h(t)\|_0 \leq \|\eta_h(0)\|_0 + \int_0^T \| \dot \eta_h(s) \|_0\,ds. 323 | \end{equation*} 324 | Bounding each of the terms appearing gives the remaining estimates: 325 | \begin{tightalign*} 326 | \| \eta_h(0) \|_0 &= \|\Pi_h u_0 - R_h u_0 \|_0 \leq \|u_0 - \Pi_h\|_0 + \| u_0 - R_h u_0\|\\ 327 | \| \dot \xi_h \|_0 &= \| \dot u - R_h \dot u \|. 328 | \end{tightalign*} 329 | The resulting estimate comes from the convergence rate obtained from the Ritz projector. 330 | \end{proof} 331 | \end{theorem} 332 | \paragraph{The fully-discrete problem} 333 | We finally derive a convergence estimate for the fully-discrete problem. As for finite differences, we could choose a forward, backward or centered difference scheme, and we will choose an implicit scheme. Let $M\in \N$ be the number of time steps with $\Delta t = T/M$, and $t_n = n\Delta t$ with $1\leq n\leq M$, such that $u(t_n)\approx u^n$. Recall the backward difference scheme 334 | \begin{equation}\label{eq:discrete-un} 335 | D^-_{\Delta t}u^n = \frac{u^{n+1}-u^n}{\Delta t}, 336 | \end{equation} 337 | which we will simply call $D_{\Delta t} u^n$. Substituting this difference in our continuous weak formulation~\ref{eq:weak-form-continuous-parabolic-IBVP} results in 338 | \begin{equation} 339 | (D_{\Delta t}u_h^n, v_h) + a(u_h^n, v_h) = \langle f^n, v_h\rangle \qquad \forall v_h\in V_h, \forall n\in\{1,\dots,M\}. 340 | \end{equation} 341 | It is easy to prove the existence and uniqueness to this equation using the Lax-Milgram lemma. To derive a bound for the error, we bound the error of each time step. At time $t_n$, we have the error equation 342 | \begin{equation}\label{eq:error-eq-fully-discrete} 343 | e_h^n \coloneqq u(t_n) - u_h^n = \underbrace{u(t_n) - \Pi_h u(t_n)}_{\xi_h^n} + \underbrace{\Pi_h u(t_n) - u_h^n}_{\eta_h^n} = \xi_h^n + \eta_h^n, 344 | \end{equation} 345 | where $\xi_h^n$ is the projection error and $\eta_h^n$ is the consistency error. Note that 346 | \begin{tightalign*} 347 | \dot u(t_n) - D_{\Delta t}u_h^n &= \dot u(t_n) - \Pi_h D_{\Delta t} u(t_n) + \Pi_h D_{\Delta t} u(t_n) - D_{\Delta t} u_h^n \tag{adding zero}\\ 348 | &= \dot u(t_n) - D_{\Delta t} u(t_n) + D_{\Delta t}(\xi_h^n + \eta_h^n) \tag{linearity of $D_{\Delta t}$}, 349 | \end{tightalign*} 350 | and replacing back in the error equation~\ref{eq:error-eq-fully-discrete} we get 351 | \begin{equation} 352 | (\dot u(t_n) - D_{\Delta t}u(t_n) + D_{\Delta t}\xi_h^n + D_{\Delta t}\eta_h^n, v_h) + a(\xi_h^n + \eta_h^n, v_h) = 0\qquad \forall v_h\in V_h. 353 | \end{equation} 354 | Choosing $\Pi_h$ as the Ritz projector, then we have $a(\xi_h^n, v_h) = 0$. Reordering, we can now bound as 355 | \begin{equation}\label{eq:semi-discrete-eta} 356 | (D_{\Delta t} \eta_h^n, v_h) + a(\eta_h^n, v_h) = \underbrace{-(\dot{u}(t_n) - D_{\Delta t}u(t_n), v_h)}_{(w_1^n, v_h)} \underbrace{-(D_{\Delta t}\xi_h^n, v_h)}_{(w_2^n, v_h)}. 357 | \end{equation} 358 | We bound the two terms separately. For $w_2$, we note that 359 | \begin{tightalign*} 360 | w_1^n &= \dot{u}(t_n) - D_{\Delta t}u(t_n) \tag{definition of $\xi_h^n$}\\ 361 | &= \int_{t_{n-1}}^{t^n} \frac{\dot{u}(t_n) - \dot{u}(s)}{\Delta t} ds \tag{$\dot{u}(t)$ is constant in $[t_{n-1}, t_n]$}\\ 362 | &= \int_{t_{n-1}}^{t^n} \frac{1}{\Delta t} \int_s^{t_n} \ddot{u}(w)dwds\\ 363 | &\leq \int_{t_{n-1}}^{t^n} \frac{1}{\Delta t} \left[\underbrace{(t-s)}_{\leq \Delta t}\sup_{w\in[t_{n-1},t_n]}|\ddot{u}(w)|\right] ds\\ 364 | &\leq \Delta t \|\ddot u\|_\infty, 365 | \end{tightalign*} 366 | and 367 | \begin{tightalign*} 368 | w_2^n = D_{\Delta t}\xi_h^n &= D_{\Delta t}(u(t_n) - \Pi_h u(t_n)) \tag{definition of $\xi_h^n$}\\ 369 | &= (I-\Pi_h) D_{\Delta t}u_h^n\\ 370 | &= (I-\Pi_h)\frac{u^n_h - u_h^{n-1}}{\Delta t}\\ 371 | &= \frac{1}{\Delta t}(I-\Pi_h) \int_{t_{n-1}}^{t^n}\dot{u}_h(s)ds\\ 372 | &\leq \frac{Ch}{\Delta t}\|\dot{u}\|_X, 373 | \end{tightalign*} 374 | Now, since $v_h\in V_h$ is arbitrary, we substitute $v_h = \eta_h^n$ in~\eqref{eq:semi-discrete-eta} and obtain 375 | \begin{equation}\label{eq:fully-discrete-eta-bnd} 376 | (D_{\Delta t} \eta_h^n, \eta_h^n) + a(\eta_h^n, \eta_h^n) = (w_1^n, \eta_h^n) + (w_2^n, \eta_h^n). 377 | \end{equation} 378 | The first term can be bounded via Young's inequality: 379 | \begin{tightalign*} 380 | (D_{\Delta t} \eta_h^n, \eta_h^n) = \left(\frac{\eta_h^{n-1}-\eta_h^n}{\Delta t}, \eta_h^n\right) &= \frac{(\eta_h^n, \eta_h^n}{\Delta t} - \frac{(\eta_h^{n}, \eta_h^{n-1})}{\Delta t}\\ 381 | &\geq \frac{\|\dot\eta_h^n\|^2}{\Delta t} - \frac{1}{2\Delta t}\left(\|\dot\eta_h^n\|^2 + \|\eta_h^{n-1}\|^2\right) \tag{Young} \\ %! what choice was done here? 382 | &= \frac{1}{2}\left(\frac{\|\dot\eta_h^n\|^2 - \|\dot\eta_h^{n-1}\|^2}{\Delta t}\right), 383 | \end{tightalign*} 384 | and substituting in~\eqref{eq:fully-discrete-eta-bnd} we get 385 | \begin{equation} 386 | \frac{1}{\Delta t}\left(\|\eta_h^n\|^2 + (\eta_h^n, \eta_h^{n-1})\right) + \underbrace{\alpha\|\eta_h^n\|^2}_{\geq 0} \leq (w_1^n + w_2^n, \eta_h^n). 387 | \end{equation} 388 | Dropping the coercive term, and using $ab\geq \|a\|\|b\|$ and the Cauchy-Schwarz inequality on the right hand side, we obtain 389 | \begin{equation} 390 | \frac{1}{\Delta t}\left(\|\eta_h^n\|^2 - \|\eta_h^n\|\|\eta_h^{n-1}\|\right) \leq \|w_1^n+w_2^n\|\|\eta_h^n\|, 391 | \end{equation} 392 | and dividing by $\|\eta_h^n\|/\Delta t$ and adding $\|\eta_h^{n-1}\|$ on both sides, we get the bound 393 | \begin{equation} 394 | \|\eta_h^n\|\leq \|\eta_h^{n-1}\| + \Delta t\|w_1^n+w_2^n\|. 395 | \end{equation} 396 | Summing over $n\in\{1,\dots,M\}$, and telescoping on the left hand side, we get 397 | \begin{tightalign*} 398 | \|\eta_h^M\| &\leq \|\eta_h^0\| + \Delta t \sum_{n=1}^M \left(\|w_1^n+ w_2^n\|_0\right)\\ 399 | &\leq \|\eta_h^0\| + \Delta t \sum_{n=1}^M \left(\|w_1^n\|_0 + \|w_2^n\|_0\right)\tag{triangle inequality}\\ 400 | &= \|\eta_h^0\| + \Delta t \sum_{n=1}^M \|w_1^n\|_0 + \Delta t \sum_{n=1}^M \|w_2^n\|_0\\ 401 | &\leq \|\eta_h^0\| + \Delta t \sum_{n=1}^M \frac{1}{\Delta t} \int_{t_{n-1}}^{t_n} \|w_1^n\|_0 ds + \Delta t \sum_{n=1}^M \frac{1}{\Delta t} \int_{t_{n-1}}^{t_n} \|w_2^n\|_0 ds\\ 402 | &= \|\eta_h^0\| + \int_{0}^{T} \|w_1^n\|_0 ds + \int_{0}^{T} \|w_2^n\|_0 ds\\ 403 | &\leq \|\eta_h^0\| + (I-\Pi_h)\|\dot u\|_{L^1} + \Delta t \int_0^T \|\ddot{u}\| ds\\ 404 | &\leq \|\eta_h^0\| + Ch\|\dot u\|_{L^1} + \Delta t \int_0^T \|\ddot{u}\| ds. %! \Delta t missing in first term?! 405 | \end{tightalign*} 406 | Thus, we got the final bound for the fully-discrete problem 407 | \begin{equation}\label{eq:spacetime-estimate} 408 | \|\eta_h^M\| \leq \|\eta_h^0\| + C\Delta t \|\ddot u\|_{L^1(L^2)} + Ch \|\dot u\|_{L^1(L^2)}. 409 | \end{equation} 410 | Here, we can often take $\Delta t = h$, which allows us to control both terms linearly in $h$ and ensure the consistency of the method. Finally, due to the convergence rate of the Ritz projector we can immediately bound the norm of $\xi_h^n$, and thus we can conclude the convergence $\|e_h\|$ by summing the errors over $n$. 411 | 412 | \example{ 413 | One common example is the heat equation, stated as follows: Find $u$ in $L^2(0,T; H_0^1(\Omega))$ such that 414 | $$\langle \dot u, v\rangle + a(u,v) = 0 \qquad\forall v \in H_0^1(\Omega),$$ 415 | where $a(u,v)\coloneqq (\grad u, \grad v)$. The previous developments show that we can approximate this problem using the k--th order FE space 416 | $$ V_h = \{ u_h \in C(\Omega): u_h|_T \in \mathbb{P}_k\}, $$ 417 | using a backward Euler discretization in time, which yields the discrete system of finding $u^n_h$ in $V_h$ 418 | $$ \frac{1}{\Delta t}(u_h^n, v_h) + a(u^n_h, v_h) = F(v_h)+ \frac{1}{\Delta t}(u_h^{n-1}, v_h) \qquad \forall v_h\in V_h,$$ 419 | for given $u_h^0$. In fact, we obtain from~\eqref{eq:spacetime-estimate} the following convergence estimate: 420 | $$ \|u_h^n - u(t^n) \|_0 \leq C(h^k + \Delta t) . $$ 421 | A common choice that stems from this estimate is to use $\Delta t = O(h^k)$. 422 | } 423 | -------------------------------------------------------------------------------- /chapters/weak-forms-galerkin.tex: -------------------------------------------------------------------------------- 1 | As we reviewed in Chapter~\ref{chapter:finite-differences}, one can derive theoretical convergence and stability guarantees for the finite difference method, but its applicability is heavily restricted by the geometry discretization, which is often limited to regular grids in rectangular domains in $\R^n$. In this chapter, we seek to (i) formally define the conditions for well-posedness of abstract and elliptic problems, (ii) introduce Galerkin methods as the standard, finite-dimensional approximation methods, and (iii) introduce approximation of PDEs via weak (or variational) formulations that are suitable for Galerkin methods. This presentation partially follows~\cite{ern2004theory}, and will be instrumental for the analysis of Finite Element Methods, which we present in Chapter~\ref{chapter:fem}. 2 | 3 | \section{Well-posedness of abstract problems} 4 | Let $W$ and $V$ be normed vector spaces with norms $\|\cdot\|_W$ and $\|\cdot\|_V$, respectively. It is sufficient for this analysis to assume that $V$ is a reflexive Banach space and $W$ is a Banach space, but in many applications they are Hilbert spaces. We define the following abstract and continuous problem: find $u\in W$ such that 5 | \begin{equation}\label{eq:well-posedness-abstract-problem} 6 | \begin{aligned} 7 | a(u,v) = f(v),\quad \forall v\in V, 8 | \end{aligned} 9 | \end{equation} 10 | where $a$ is a continuous bilinear form on $W\times V$, i.e. $a\in \mathcal{L}(W\times V; \R)$ (and thus $a$ is bounded), and $f\in V'$ is a continuous linear form on $V$. Recall that we are using the shorthand notation $f(v)=\langle f, v\rangle_{V',V}$ for the duality pairing. 11 | 12 | The notion that problem~\ref{eq:well-posedness-abstract-problem} has a \emph{unique solution} and that it is continuous with respect to $f$ is called \emph{well-posedness}, and is the fundamental property that one seeks to prove to consider proposing a discrete method to solve it. This definition of well-posedness is standard and was proposed by Hadamard. 13 | \begin{definition}[Well-posedness in the sense of Hadamard] 14 | Problem~\ref{eq:well-posedness-abstract-problem} is said to be \emph{well-posed} if it admits exactly one solution, and if the following a priori estimate holds: 15 | \begin{equation} 16 | \exists c>0, \forall f\in V',\quad \|u\|_W \leq c\|f\|_{V'}. 17 | \end{equation} 18 | \end{definition} 19 | 20 | In our applications, the bilinear form $a$ and the linear form $f$ result from deriving a \emph{weak formulation} of a boundary value problem, i.e. a PDE with a set of boundary conditions. In the following section, we will derive such formulations for a number of PDEs. 21 | 22 | \section{Weak formulations}\label{sec:weak-formulations} 23 | A weak formulation refers to an integral form of a PDE, understood distributionally. This is typically a systematic procedure that should not be too difficult, and it helps in revealing what are the adequate boundary conditions for a given problem. The main tool for this will be the integration by parts formulas. 24 | 25 | Our test problem will be the Laplace problem, given by the $-\Delta$ operator. The minus sign will be better justified in the following section. Consider then the problem of finding $u$ such that 26 | \begin{equation} 27 | -\Delta u = f \quad \tin\Omega. 28 | \end{equation} 29 | Define an arbitrary smooth function $v$, then integration by parts yields 30 | \begin{equation}\label{eq:ibp-poisson} 31 | - \int_\Omega \Delta u v\,dx = -\int_{\partial\Omega}\gamma_0 v \gamma_N \grad u \,ds + \int_\Omega \grad u \cdot \grad v\,dx 32 | \end{equation} 33 | for all $v$. This function is typically called a \emph{test function}. The surface form suggest the boundary conditions: 34 | \begin{equation} 35 | \int_{\partial\Omega}\underbrace{\gamma_0 v}_\text{Dirichlet BC} \underbrace{\gamma_N \grad u}_\text{Neumann BC} \,ds, 36 | \end{equation} 37 | so that we can have boundary conditions on the function itself 38 | \begin{equation*} 39 | u = g, 40 | \end{equation*} 41 | or on its normal derivative 42 | \begin{equation*} 43 | \grad u \cdot \vec n = h. 44 | \end{equation*} 45 | This can be combined, so that for a given partition of the boundary into two disjoint sets $\Gamma_D$ and $\Gamma_N$ such that $\overline{\partial\Omega} = \overline{\Gamma_D}\cup\overline{\Gamma_N}$, one can have a Dirichlet boundary condition on $\Gamma_D$ and a Neumann boundary condition on $\Gamma_N$. For this type of boundary condition, one must define a solution space given by 46 | \begin{equation} 47 | V_g = \{v \in H^1(\Omega): \gamma_0 v = g \ton\Gamma_D\}, 48 | \end{equation} 49 | but let us focus first on spaces with null Dirichlet boundary condition ($V_0$). In this case, the boundary conditions will give 50 | \begin{equation*} 51 | \int_{\partial\Omega}\gamma_0 v \gamma_N \grad u\,ds = \int_{\gamma_N} h \gamma_0 v\,ds, 52 | \end{equation*} 53 | and thus the integral form of the equation will be given by 54 | \begin{equation*} 55 | -\int_\Omega\Delta u v\,dx = -\int_{\Gamma_N}v h\,ds + \int_\Omega \grad u \cdot \grad v\,dx = \int_\Omega f v\,dx, 56 | \end{equation*} 57 | for all smooth $v$. Now, we note that (i) this formulation is well defined for $u,v$ in $H^1$ (and thus can be extended to hold for all $v$ in $H^1$ by density as long as $v$ satisfies the Dirichlet boundary conditions), (ii) the trace operator $\gamma_0$ has been omitted from the surface integral for convenience, and (iii) that the Dirichlet boundary condition does not appear anywhere in the formulation. This justifies naming Dirichlet boundary conditions \emph{essential}, and Neumann boundary conditions \emph{natural}. The \emph{weak formulation} of the problem thus refers to the following statement: find $u$ in $V_0$ such that 58 | \begin{equation}\label{eq:weak-form-poisson} 59 | (\grad u, \grad v)_{0,\Omega} = \langle f, v\rangle + \langle h, v\rangle \qquad \forall v\in V_0, 60 | \end{equation} 61 | for given functions $f$ in $V_0'$ and $g$ in $(\gamma_0 V_0)'$. Note the following: 62 | \begin{itemize} 63 | \item The space of the solution and the test functions is the same. This is not mandatory, but it is common and can be better motivated by interpreting the Laplace problem as the first order equations related to the following minimization problem: 64 | \begin{equation*} 65 | \min_{v \in V_0} \int_\Omega |\grad v|^2\,dx . 66 | \end{equation*} 67 | Then, one simply infers the spaces of each function from the definition of the Gâteaux derivative. 68 | \item The solution $u$ was formulated in a space without boundary condition. This is important because the regularity theory will depend on the solution space being a Hilbert space, and the space $V_g$ is not even a vector space as it is not closed under addition. This can be solved by defining adequate \emph{lifting} operators, i.e. a function $G$ in $H^1(\Omega)$ such that $\gamma_0 G = g$ that allows us to write $u$ in $V_g$ as 69 | \begin{equation*} 70 | u = u_0 + G, 71 | \end{equation*} 72 | where $u_0$ belongs to $V_0$. We can then rewrite the problem in $V_g$ as a problem in $V_0$ (and I encourage the reader to do this procedure at least once in their life). The existence of a lifting function in this case is given by the surjectivity of the Dirichlet trace, but it can be tricky in other contexts. This is also tricky in nonlinear problems, which justifies that nonlinear problems are typically studied with homogeneous boundary conditions. 73 | \item We note that the Laplacian is now being interpreted as a \emph{distribution}, and thus the strong problem (including the boundary conditions) yields the definition of the \emph{action} of the distribution. In particular, this means that the action of the distribution naturally changes with the boundary conditions. This observation is fundamental to understand Discontinuous Galerkin methods, or other formulations defined on broken spaces, i.e. spaces that allow for discontinuities. 74 | \end{itemize} 75 | 76 | \section{Elliptic problems and the Lax-Milgram lemma} 77 | A broad class of PDEs where we can prove existence and uniqueness of solutions are \emph{elliptic} problems, whose weak formulations involve elliptic forms. 78 | \begin{definition}[Elliptic forms]\label{def:elliptic-form} 79 | A bilinear form $a(\cdot, \cdot)$ defined on a Hilbert space $X$ is said to be \emph{elliptic} if there exists a constant $\alpha>0$ such that 80 | \begin{equation}\label{eq:elliptic-form} 81 | a(x, x) \geq \alpha \| x \|^2_X \qquad \forall x\in X. 82 | \end{equation} 83 | \end{definition} 84 | 85 | The ellipticity property~\eqref{eq:elliptic-form} is the basis of the Lax-Milgram lemma, which gives sufficient conditions for the existence and uniqueness of solutions of elliptic problems. 86 | \begin{lemma}[Lax-Milgram]\label{lemma:lax-milgram} 87 | Consider a bounded bilinear form $a: H\times H\to \R$ defined on a Hilbert space $H$ that is elliptic with constants $C$ and $\alpha$ respectively, and a linear functional $f\in H'$. Then, problem~\eqref{eq:well-posedness-abstract-problem} is well-posed, i.e. there exists a unique solution $u\in H$ such that $a(u,v)=f(v)$ for all $v\in H'$, and it satisfies the \emph{a priori estimate} 88 | \begin{equation}\label{eq:lax-milgram-a-priori} 89 | \| u\|_H \leq \frac 1 \alpha \| f \|_{H'} . 90 | \end{equation} 91 | \end{lemma} 92 | 93 | Before providing a proof, we note that every continuous bilinear form $a:H\times H\to \R$ induces an operator $A:H\to H'$ given by 94 | \begin{equation}\label{eq:form-induced-by-matrix} 95 | (Au)[v] = a(u,v), 96 | \end{equation} 97 | which one could also write as $Au = a(u, \cdot)$. Naturally, the bilinear form $a$ is bounded if and only if the operator $A$ is bounded. 98 | 99 | \begin{proof} 100 | It will be seen further ahead that this can be easily proved using the inf-sup conditions. Still, we present a more elementary proof that uses only the properties of the bilinear form and a fixed point argument. Consider $\rho>0$ and the fixed-point map $T:H\to H$ given by 101 | \begin{equation*} 102 | T(u) = u - \rho \mathcal R^{-1}\circ (Au - F), 103 | \end{equation*} 104 | where it can be seen that $T$ is linear, and $\mathcal R$ is the Riesz map between $H$ and $H'$. Now, we look for $\rho$ such that $T$ is a contraction, which we do simply by hand. Consider thus two functions $u,v$ in $H$, then: 105 | \begin{tightalign*} 106 | \| T(u) - T(v)\|_H^2 &= \|T(u - v) \|_H^2 \\ 107 | &= (u-v, u-v)_H - 2\rho(u-v, \mathcal R^{-1}\circ A(u-v))_H\\ 108 | &\phantom{= } + \rho^2(\mathcal R^{-1}\circ A(u-v), \mathcal R^{-1}\circ A(u-v))_H \\ 109 | &= \|u-v\|_H^2 - 2\rho\langle A(u-v), u-v\rangle_{H'\times H} + \rho^2 \| \mathcal R^{-1} \circ A(u-v)\|_H^2. 110 | \end{tightalign*} 111 | We bound the second and third terms as follows: 112 | \begin{tightalign*} 113 | \langle A(u-v), u-v\rangle &= a(u-v, u-v) \tag{by definition of $A$}\\ 114 | &\geq \alpha \| u-v\|_H^2, \tag{ellipticity} 115 | \end{tightalign*} 116 | \begin{tightalign*} 117 | \| \mathcal R^{-1} \circ A(u-v) \|_H &= \| A(u-v) \|_{H'} \tag{Riesz isometry}\\ 118 | &\leq C \| u-v \|_H. \tag{continuity} 119 | \end{tightalign*} 120 | Plugging this into our previous estimate we get 121 | \begin{equation*} 122 | \| Tu - Tv \|_H \leq (1 - 2\rho \alpha + \rho^2 C^2)^{1/2}\| u-v \|_H , 123 | \end{equation*} 124 | which shows that $T$ is a contraction whenever $\rho\in (0,\frac{2\alpha}{C^2})$. Stability follows naturally from the properties of $a$: 125 | \begin{tightalign*} 126 | \alpha \| u \|^2 & \leq a(u,u) \tag{ellipticity}\\ 127 | & = f(u) \\ 128 | & \leq \| f \|_{H'} \|u \|_H, 129 | \end{tightalign*} 130 | which shows the desired stability estimate: 131 | \begin{equation*} 132 | \| u \|_H \leq \frac{1}{\alpha} \| f \|_{H'}. 133 | \end{equation*} 134 | \end{proof} 135 | 136 | In the case that $a$ is symmetric and non-negative, problem~\ref{eq:well-posedness-abstract-problem} is equivalent to the optimization problem 137 | \begin{equation*} 138 | \min_{u\in V} J(u) \coloneqq \min_{u\in V} \left\{\frac{1}{2}a(u,u) - f(u)\right\}, 139 | \end{equation*} 140 | and the Lax-Milgram lemma implies that there exists a unique minimizer. In this sense, the ellipticity of $a$ translates to the strong convexity of $J$, which represents an \emph{energy functional}. 141 | 142 | The generalization of the Lax-Milgram lemma for the setting where $W$ is a Banach space and $V$ is a reflexive Banach space is covered in Chapter~\ref{chapter:nonlinear}, where we state the \emph{generalized Lax-Milgram lemma} and the \emph{inf-sup condition}. 143 | 144 | \section{Galerkin methods}\label{sec:galerkin} 145 | Instead of discretizing the differential operator, as one would do in the case of finite differences to obtain discrete derivatives, one can consider discrete functional spaces, with the idea that the discrete space somehow converges to the continuous space. This is known as a \emph{Galerkin scheme} or \emph{Galerkin method}. 146 | 147 | \paragraph{The discrete Galerkin setting} 148 | Consider thus an abstract differential problem given by finding $u$ in $W$ such that 149 | \begin{equation} 150 | a(u, v) = f(v), \quad \forall v \in V, 151 | \end{equation} 152 | such that the hypotheses the Lax-Milgram lemma~\ref{lemma:lax-milgram} hold. We can consider a discrete \emph{solution (or trial) space} $W_h$ and a discrete \emph{test space} $V_h$ that approximate $W$ and $V$, where the subindex $h$ refers to the characteristic size of the elements of a discrete mesh of the domain $\Omega$. and define a \emph{Petrov-Galerkin method (or scheme)} as the following discrete problem: find $u_h \in W_h$ such that 153 | \begin{equation}\label{eq:galerkinscheme} 154 | a_h(u_h, v_h) = f_h(v_h) \qquad \forall v_h \in V_h, 155 | \end{equation} 156 | where $a_h$ and $f_h$ are approximations of $a$ and $f$, respectively. 157 | We assume that both spaces $W_h$ and $V_h$ are equipped with some norms $\|\cdot\|_{W_h}$ and $\|\cdot\|_{V_h}$. In most references, the particular case of~\ref{eq:galerkinscheme} when $W_h = V_h$ is known as a \emph{standard Galerkin}, \emph{Bubnov-Galerkin} or simply \emph{Galerkin method}. 158 | 159 | We note that Galerkin methods are simply linear systems. To see this, consider~\ref{eq:galerkinscheme} and fix $M=\dim W_h$, $N=\dim V_h$. We can then fix $\{\psi_1,\dots,\psi_M\}$ a basis of $W_h$ and $\{\varphi_1,\dots,\varphi_N\}$ a basis of $V_h$. The expansion of $u_h\in W_h$ reads 160 | \begin{equation} 161 | u_h = \sum_{i=1}^M U_i \psi_i, 162 | \end{equation} 163 | where we denote the coordinate vector of $u_h$ as $\vec U = (U_i)_i\in\R^M$ with respect to the basis $\{\psi_1,\dots,\psi_M\}$. Denote the matrix $\ten A\in\R^{N\times M}$ and the vector $\vec F \in \R^N$ component-wise as 164 | \begin{equation} 165 | A_{ij} \coloneqq a_h(\psi_j, \varphi_i),\quad F_i\coloneqq f_h(\varphi_i), 166 | \end{equation} 167 | and one can readily verify that $u_h$ solves~\ref{eq:galerkinscheme} if and only if $\ten A\vec U =\vec F$. When working with a computational implementation of a Galerkin method, we have to take advantage of the structure of $\ten A$ to propose fast methods for solving the linear system on $\vec U$. 168 | 169 | The choice of the $W_h$ and $V_h$ depends on the scheme used, and one may or may not choose them as subsets of the corresponding continuous function spaces. For instance, when approximating $W=C^2$ functions in 1D, one could choose the space $W_h$ as the space of piecewise linear functions, where $W_h\not\subset W$, or as the space of cubic splines, where $W_h\subset W$. This property is known as \emph{conformity}. 170 | \begin{definition}[Conformity] 171 | An approximation setting (such as a Galerkin scheme) is said to be \emph{conformal} if $W_h\subset W$ and $V_h\subset V$, and is said to be \emph{non-conformal} otherwise. 172 | \end{definition} 173 | 174 | Naturally, we would expect the continuous solution $u\in W$ to be a solution 175 | \begin{equation*} 176 | a_h(u, v_h) = f_h(v_h),\quad \forall v_h\in V_h, 177 | \end{equation*} 178 | where $a_h$ is appropriately extended to handle $u$. In this case, we say that our method is \emph{consistent}, otherwise it is \emph{non-consistent}. This property implies an additional property: by subtracting equations 179 | \begin{tightalign*} 180 | a_h(u, v_h) &= f_h(v_h),\quad \forall v_h\in V_h,\\ 181 | a_h(u_h, v_h) &= f_h(v_h),\quad \forall v_h\in V_h, 182 | \end{tightalign*} 183 | and by the bilinearity of $a_h$ we preserve in the approximation $a_h\approx a$, we obtain the \emph{Galerkin orthogonality} property 184 | \begin{equation}\label{eq:galerkin-orthogonality} 185 | a_h(u-u_h, v_h) = 0,\quad \forall v_h\in V_h. 186 | \end{equation} 187 | 188 | Most of the time we will use conformal, consistent, standard Galerkin methods, where $W_h=V_h$ and $V_h\subset V$. When $a$ is elliptic, it is easy to verify that all the hypotheses of the Lax-Milgram lemma are satisfied, and thus the Galerkin scheme is invertible and we obtain a discrete a priori estimate 189 | \begin{equation*} 190 | \|u_h\|_V \leq \frac{1}{\alpha}\|f\|_{V'}. 191 | \end{equation*} 192 | One can further prove that for elliptic problems, the matrix $\ten{A}$ associated to the Galerkin scheme is positive-definite, and if $a$ is symmetric, then $\ten{A}$ is symmetric as well. The general case, when $a$ is not necessarily elliptic, is covered in~\ref{sec:inf-sup}. 193 | 194 | \paragraph{Error analysis} 195 | The natural question is whether the discrete solution $u_h\in W_h$ converges to the continuous solution $u\in W$, which is studied through the \emph{error equation}. We define the \emph{approximation error} $e_h$ as 196 | \begin{equation} 197 | e_h\coloneqq u - u_h. 198 | \end{equation} 199 | 200 | We can use the Galerkin orthogonality~\ref{eq:galerkin-orthogonality} to compute the error estimate for consistent, conformal approximations of elliptic problems. This estimate is known as \emph{Céa's estimate}. 201 | \begin{lemma}[Céa's estimate]\label{lemma:cea-estimate} 202 | Consider a consistent, conformal approximation of~\ref{eq:galerkinscheme}, i.e. with $W_h=V_h$ and $W=V$. Assume that the bilinear form $a$ is elliptic with ellipticity constant $\alpha$ and continuity constant $C$. Then, we have the error estimate 203 | \begin{equation} 204 | \| u - u_h \|_V \leq \frac C \alpha \inf_{v_h\in V_h} \|u - v_h\|_V. 205 | \end{equation} 206 | \begin{proof} 207 | Let $z_h\in V_h$ be an arbitrary test function. We proceed directly: 208 | \begin{tightalign*} 209 | \alpha \| e_h \|_V^2 &\leq a(e_h, e_h) \tag{$a$ elliptic}\\ 210 | &= a(e_h, u - z_h) \tag{Galerkin orthogonality} \\ 211 | &\leq C\|e_h\|_V \|u - z_h\|_V \tag{$a$ continuous}, 212 | \end{tightalign*} 213 | where one obtains that for all $z_h$ it holds that 214 | \begin{equation} 215 | \| e_h \|_V \leq \frac C \alpha \|u - z_h\|_V. 216 | \end{equation} 217 | Taking the infimum over $z_h$ we get the desired bound. 218 | \end{proof} 219 | \end{lemma} 220 | 221 | This inequality can reveal many things. For example, if the number $C/\alpha$ is very big, it can hint on a very wide gap between the optimal solution (i.e. the projection) and the discrete one computed from the space $V_h$. This estimate can be sharpened when $a$ is symmetric via an energy norm, and it extended to the non-elliptic case, as we will study in~\ref{sec:inf-sup}, as well as the non-consistent and non-conformal case, see~\cite[Sect. 2.3.2]{ern2004theory}. A more precise characterization of the approximation properties of a space can be given by the \emph{Kolmogorov width}, which has been studied in~\cite{evans2009n}. 222 | 223 | \paragraph{The Aubin-Nitsche lemma} We can derive an error estimate in a weaker norm than Céa's estimate. For simplicity, we consider the conformal, consistent, standard Galerkin variant of~\ref{eq:galerkinscheme}, where $W_h=V_h$ and $W=V$. We assume that our continuous and discrete problems are well-posed, as well as two additional assumptions: 224 | \begin{enumerate} 225 | \item There exists a Hilbert space $H$ into which $V$ can be continuously embedded, i.e. for $\|v\|_H \leq c\|v\|_V$ for all $v\in V$. Assume that $H$ is equipped with a continuous, symmetric, positive bilinear form $\ell(\cdot,\cdot)$, which induces a seminorm $|\cdot|_H \coloneqq\sqrt{\ell(\cdot,\cdot)}$. Further, assume that there exists a Banach space $Z\subset V$ and a stability constant $c_S>0$ such that for all $g\in H$, the function $\sigma(g)\in V$ solves the auxiliary problem 226 | \begin{equation*} 227 | a(v, \sigma(g)) = \ell(g, v),\quad \forall v\in V, 228 | \end{equation*} 229 | and satisfies the a priori estimate $\|\sigma(g)\|_Z \leq c_S |g|_H$. 230 | \item There exists an interpolation constant $c_h>0$ such that for all mesh sizes $h$, 231 | \begin{equation*} 232 | \inf_{v_h\in V_h} \|v-v_h\|_V \leq c_h h \|v\|_Z. 233 | \end{equation*} 234 | \end{enumerate} 235 | 236 | With these two assumptions, we can derive an error estimate in the seminorm $|\cdot|_H$. 237 | \begin{lemma}[Aubin-Nitsche lemma] 238 | Under the above assumptions, for all mesh sizes $h$ we have 239 | \begin{equation*} 240 | |u-u_h|_H \leq c h \|u-u_h\|_V, 241 | \end{equation*} 242 | where $c=c_hc_S \|a\|_{W\times V}$. 243 | \begin{proof} 244 | Setting the error $e_h = u-u_h$, we can see that 245 | \begin{equation*} 246 | |e_h|_H = \sup_{g\in H}\frac{\ell(g,e_h)}{|g|_H}=\sup_{g\in H}\frac{a(e_h, \sigma(g))}{|g|_H}. 247 | \end{equation*} 248 | Galerkin orthogonality implies that $a(e_h,\sigma(g)) = a(e_h, \sigma(g)-v_h)$ for all $v_h\in V_h$, and thus 249 | \begin{tightalign*} 250 | a(e_h, \sigma(g)) &\leq \|a\|_{W\times V} \|e_h\|_V \inf_{v_h\in V_h} \|\sigma(g) - v_h\|_V\\ 251 | &\leq \|a\|_{W\times V} \|e_h\|_V c_h h \|\sigma(g)\|_Z\tag{first assumption}\\ 252 | &\leq \|a\|_{W\times V} \|e_h\|_V c_h h c_S |g|_H, \tag{second assumption} 253 | \end{tightalign*} 254 | which yields the estimate we wanted. 255 | \end{proof} 256 | \end{lemma} 257 | 258 | \paragraph{Examples} 259 | \paragraph{The Poisson problem} Consider $f$ in $H^{-1}(\Omega)$ and $g$ in $H^{1/2}(\Gamma)$ with $\Gamma\coloneqq \partial\Omega$. The Poisson problem in strong form is given as the following boundary value problem: 260 | \begin{equation*} 261 | \begin{aligned} 262 | -\Delta u &= f &&\tin\Omega\\ 263 | \gamma_0 u &= g &&\ton\Gamma. 264 | \end{aligned} 265 | \end{equation*} 266 | Note that the strong form must be understood in the distributional sense, i.e. as an equation in $H^{-1}(\Omega)$. To derive the weak formulation, consider a function $v$ in $H_0^1(\Omega)$, then using the boundary conditions we obtain that 267 | \begin{equation} 268 | -\langle \Delta u,v\rangle = (\grad u, \grad v), 269 | \end{equation} 270 | where $(\cdot, \cdot)$ is the $L^2(\Omega)$ product. Thus the weak formulation reads: find $u$ in $H_0^1(\Omega)$ such that 271 | \begin{equation} 272 | \int_\Omega \grad u\cdot \grad v\,dx = \int_\Omega fv\, dx \qquad \forall v\in H_0^1(\Omega), 273 | \end{equation} 274 | or more compactly, $(\nabla u, \nabla v) = (f, v)$. This problem can be shown to be well-posed using Lax-Milgram's lemma and the Poincaré inequality. A good exercise is to extend the proof to the case of non-homogeneous Dirichlet boundary conditions. 275 | 276 | In the case of having a boundary condition defined only on a portion $\Gamma_D$ of the boundary, the formulation changes, because (i) we need further information regarding the Neumann trace on the complement of the boundary, (ii) the test space looks different. In particular, we define the solution space given by 277 | \begin{equation}\label{eq:def-V0} 278 | V_0 = \{v\in H^1(\Omega): \quad v = 0 \ton \Gamma_D\}, 279 | \end{equation} 280 | which using the generalized Poincaré inequality can be shown to still satisfy an ellipticity estimate. 281 | 282 | \paragraph{The pure Neumann problem} In general, having Neumann boundary conditions is problematic for two reasons: (i) it results in a \emph{data compatibility} condition, and (ii) it results in having a non-trivial kernel in the problem. The problem in general reads: find $u$ in $H^1(\Omega)$ such that 283 | \begin{equation} 284 | \begin{aligned} 285 | -\Delta u &= f && \tin\Omega ,\\ 286 | \grad u \cdot \vec n &= h && \ton\partial\Omega. 287 | \end{aligned} 288 | \end{equation} 289 | The weak formulation is 290 | \begin{equation} 291 | (\grad u, \grad v) = (f, v) \qquad \forall v\in H^1(\Omega), 292 | \end{equation} 293 | where it is easy to see that if $u$ is a solution, then $u+c$ is also a solution for all $c\in \R$. This means that the problem has a kernel, which is given by the space of constant functions, i.e. $\spanned(\{1\})$. The other problem is that, when one considers a test function in the kernel of the problem, this yields the following: 294 | \begin{equation} 295 | (\grad u, \grad 1) = 0 = (f, 1) . 296 | \end{equation} 297 | This is a compatibility condition on the data, and it shows that having compatible data is \emph{necessary} for having a well-posed formulation. Because of these reasons, one considers a solution (and test) space that is orthogonal to the kernel: 298 | \begin{equation} 299 | V = \{u\in H^1(\Omega): \int_\Omega u \,dx = 0\}, 300 | \end{equation} 301 | where the null average condition can be seen as 302 | \begin{equation} 303 | \int_\Omega u \,dx = (u, 1)_0 = (u,1)_0 + (\grad u, \grad 1)_0 = (u, 1)_1, 304 | \end{equation} 305 | and thus the orthogonality is being considered with respect to the natural space $H^1(\Omega)$. With it, the weak formulation is given as: Consider $f$ a compatible function in $H^{-1}(\Omega)$, then find $u$ in $V$ such that 306 | \begin{equation} 307 | (\grad u, \grad v) = (f, v) \qquad \forall v\in V. 308 | \end{equation} 309 | 310 | To conclude this section, we wanted to establish that if the solution is sufficiently regular, then the weak form is equivalent to the strong form. The main tool for this is the fundamental lemma of the calculus of variations: 311 | 312 | \begin{lemma}[Fundamental lemma of the calculus of variations]\label{lemma:lema-calvar} 313 | Consider $\Omega\subset \R^d$ bounded and set $f\in L^1(\Omega)$ such that 314 | \begin{equation}\label{eq:lemma-calvar} 315 | \int_\Omega f\varphi\,dx = 0 \qquad \forall \varphi \in C_0^\infty(\Omega). 316 | \end{equation} 317 | Then, $f=0$ almost everywhere. 318 | \end{lemma} 319 | 320 | Consider now the weak formulation of the Poisson problem: Find $u$ in $V_0$ such that 321 | \begin{equation} 322 | (\grad u, \grad v) = (f, v) - (\grad u_g, \grad v) + \langle t, v\rangle_{-1/2, 1/2}, 323 | \end{equation} 324 | where $f$ and $t$ are integrable. Then, integrating by parts one obtains 325 | \begin{equation}\label{eq:ibp-nonhomogeneous-poisson} 326 | (-\Delta (u+u_g) - f, v)+\langle \grad u \cdot \vec n, v\rangle = \langle t, v\rangle_{-1/2,1/2}. 327 | \end{equation} 328 | If we consider test functions $v$ in $C_0^\infty(\Omega)$, then the problem reduces to 329 | \begin{equation} 330 | (-\Delta (u+u_g) - f, v) = 0, 331 | \end{equation} 332 | and if $u$ and the lifting function $u_g$ are integrable, then Lemma~\ref{lemma:lema-calvar} yields 333 | \begin{equation*} 334 | -\Delta \tilde u = f, 335 | \end{equation*} 336 | which is the strong form for the combined solution $\tilde u= u + u_g$. Substituting this back into equation~\eqref{eq:ibp-nonhomogeneous-poisson}, we obtain the weak form 337 | \begin{equation*} 338 | \langle \grad u \cdot \vec n, v\rangle = \langle t, v\rangle. 339 | \end{equation*} 340 | Using again Lemma~\ref{lemma:lema-calvar} on the subspace topology of $C_0^\infty(\Gamma_N)$, then we obtain the equation 341 | \begin{equation*} 342 | \grad u\cdot \vec n = t, 343 | \end{equation*} 344 | which holds strongly.\\ 345 | 346 | \example{ 347 | We encourage the reader to try to compute the weak formulation of the Poisson problem in mixed form. To do this, one must define the auxiliary variable $\vec \sigma \coloneqq \grad u$, so that the strong form of the problem now becomes 348 | \begin{equation*} 349 | \begin{aligned} 350 | -\dive \vec\sigma &= f && \tin\Omega, \\ 351 | \vec \sigma - \grad u &= 0 &&\tin\Omega , \\ 352 | \gamma_D u &= g && \ton\Gamma_D, \\ 353 | \gamma_N \vec\sigma &= h &&\ton\Gamma_D . 354 | \end{aligned} 355 | \end{equation*} 356 | This problem will be studied in detail further ahead. 357 | } 358 | \paragraph{Non-conforming schemes}\label{sec:non-conforming-schemes} 359 | In order to have a good approximation, it is not necessary that the scheme is conforming. Some examples of conforming schemes are most finite element methods and spectral element methods, and among non-conforming schemes are discontinuous-Galerkin (DG) finite element methods, and methods that impose boundary conditions weakly. We will study several conforming finite element methods in Chapter~\ref{chapter:fem}. 360 | 361 | \paragraph{Classical non-conforming schemes} Although conforming spaces have some nice properties, there exist some applications where the mesh and the domain boundaries may not match, or where traditional finite elements may not apply, forcing one to use other schemes such as spline-based methods, where the degrees of freedom are control points of the splines, rather than actual nodes. In these cases, we can use a non-conforming scheme. The following presentation is based on~\cite{Chouly2024}. Consider the Poisson problem with a nonhomogeneous Dirichlet boundary condition: find $u:\Omega\to \mathbb{R}$ such that 362 | \begin{equation} 363 | \begin{aligned} 364 | -\Delta u &= f &&\tin \Omega \\ 365 | \gamma_0 u &= g &&\ton \partial\Omega, 366 | \end{aligned} 367 | \end{equation} 368 | which we rewrite in weak form as follows: find $u\in H^1(\Omega)$ such that $u|_{\partial\Omega} = g$ and 369 | \begin{equation} 370 | a(u,v) = (f,v) \qquad \forall v\in H_0^1(\Omega), 371 | \end{equation} 372 | where as usual we denote $a(u,v)=(\nabla u,\nabla v)$. Let $K^h$ be a discretization of $\Omega$ with mesh size $h$, which we assume is sufficiently regular. There exist several ways to enforce the Dirichlet boundary condition, such as the \emph{penalty method} and the \emph{Nitsche method}. We outline both methods below. 373 | \begin{itemize} 374 | \item \emph{The penalty method}: at the continuous level, this method can be formulated as follows: find $u^\varepsilon\in H^1(\Omega)$ such that 375 | \begin{equation} 376 | a(u^\varepsilon, v) + \frac{1}{\varepsilon} (u^\varepsilon, v)_{\partial\Omega} = (f,v) + \frac{1}{\varepsilon} (g,v)_{\partial\Omega} \qquad \forall v\in H^1(\Omega), 377 | \end{equation} 378 | where we introduced the penalty parameter $\varepsilon>0$. When going back to the strong form, we verify that $u^\varepsilon$ satisfies the Poisson equation $-\Delta u^\varepsilon = f$ and the Robin boundary condition 379 | \begin{equation} 380 | \nabla u^\varepsilon \cdot\vec n = -\frac{1}{\varepsilon}(u^\varepsilon - g) \implies \varepsilon (\nabla u^\varepsilon) \cdot \vec n = -(u^\varepsilon - g), 381 | \end{equation} 382 | which for $\varepsilon$ small enough approximates the nonhomogeneous Dirichlet boundary condition. By the Friedrich inequality, we can show that the bilinear form in the left hand side is elliptic on $H^1(\Omega)$ and thus the problem is well-posed by the Lax-Milgram lemma. In a discrete setting, we consider $\varepsilon = \varepsilon_0 h^\lambda$ for some $\varepsilon_0 > 0$ and $\lambda\geq 0$, both independent of the mesh, and we can prove that this discrete problem is well-posed and convergent. Often, the user has to manually tune the values of $\varepsilon_0$ and $\lambda$ to achieve good convergence rates. The critical choice lies in the value of $\varepsilon_0$: if the value is too small, the conditioning of the global stiffness matrix deteriorates, since its conditioning is $\mathcal{O}(\varepsilon_0^{-1}h^{-1-\lambda})$, and if the value is too large, the Dirichlet condition is approximated poorly. 383 | 384 | \item \emph{The Nitsche method}: let $\gamma > 0$ be a positive function on $\partial\Omega$ and $\theta\in\mathbb{R}$ a fixed parameter. Integrating by parts the weak form of the Poisson problem we first get 385 | \begin{equation} 386 | a(u,v) - (\nabla u\cdot \vec n, v)_{\partial\Omega} = (f,v), 387 | \end{equation} 388 | and from the Dirichlet condition we can write 389 | \begin{equation} 390 | (u,\gamma v -\theta\nabla v\cdot \vec n)_{\partial\Omega} = (g,\gamma v - \theta \nabla v \cdot \vec n)_{\partial\Omega}. 391 | \end{equation} 392 | Adding these two equations together and rearranging, we get 393 | \begin{equation} 394 | a(u,v) - (\nabla u \cdot \vec n, v)_{\partial\Omega} - \theta (u,\nabla v\cdot \vec n) + (u,\gamma v)_{\partial\Omega} = (f,v) + (g,\gamma v - \theta \nabla v\cdot n)_{\partial\Omega}. 395 | \end{equation} 396 | Let $\zeta$ denote a piecewise constant function on the boundary, that is defined locally by the value of the diameter of every boundary facet. Taking $\gamma = \gamma_0 \zeta^{-1}$ for some $\gamma_0>0$, and recalling the trace inequality~\eqref{eq:trace-inequality}, 397 | \begin{equation} 398 | \|\nabla v_h\cdot \vec n\|^2_{-1/2,\partial\Omega} \leq c_T \|\nabla v_h\|^2_{0,\Omega}, 399 | \end{equation} 400 | we can prove that this problem is well-posed provided that 401 | \begin{equation} 402 | \frac{(1+\theta)^2 c_T}{\gamma_0}\leq 1. 403 | \end{equation} 404 | Moreover, this method is convergent in the $H^1$ norm for large enough $\gamma_0$. In the case that we expect more regularity, for $u\in H^s(\Omega)$ with $3/20$ does not depend on $\gamma_0$ provided that it is large enough, but does depend on the regularity of the mesh and on the polynomial order $k$. As expected, the value of $\gamma_0$ influences the condition number of the global stiffness matrix associated to the left hand side of this problem, and thus it must not be taken too large, but the impact of the value of $\gamma_0$ on the approximation of the Dirichlet boundary condition is much smaller than in the penalty method. 409 | \end{itemize} 410 | 411 | In practice, the penalty method is much simpler to understand and to implement, but its accuracy in some specific problems may not always be satisfactory. The Nitsche method is still simple to implement, and it constitutes a better alternative to the penalty method, where one has to tune only one numerical parameter. There exist more variants to these methods, such as the penalty-free Nitsche method and methods with Lagrange multipliers. The interested reader is referred to~\cite{Chouly2024} for more details. 412 | --------------------------------------------------------------------------------