Corrigenda Numerical Methods Nonlinear PDEs

$\def\R{\mathbb{R}}$ $\def\C{\mathbb{C}}$ $\def\O{\Omega}$ $\def\cS{\mathcal{S}}$ $\def\cT{\mathcal{T}}$ $\def\veps{\varepsilon}$ $\def\vphi{\varphi}$ $\def\hT{\widehat{T}}$ $\def\tu{\widetilde{u}}$ $\newcommand{norm}[1]{\|#1\|}$ $\newcommand{abs}[1]{|#1|}$

The following list of corrections is updated occasionally and ordered chronologically. Letters t,m,b refer top, middle, and bottom parts of a page.

Last update: September 5, 2017

Chapter 1: Introduction

Chapter 2: Analytical Background

Nr. Page(s) Correction
1 30b Def. 2.3 (b), add: for all $h\in X$
2 44t Correct title: Nichtlineare Funktionalanalysis

Chapter 3: FEM for Linear Problems

Nr. Page(s) Correction
1 56m Replace integration domain $\Gamma_D$ by $\Gamma_N$
2 81t Add command d_fac = factorial(d) and compute with this quantity (same on pages 83, 281, 318, 322, 374, 377)
3 83m Replace command diag(m_lumped_diag) by spdiags(m_lumped_diag,0,nC,nC)
4 83b Replace command diag(m_lumped_bdy_diag) by spdiags(m_lumped_bdy_diag,0,nC,nC)

Chapter 4: Concepts for Discretized Problems

Nr. Page(s) Correction
1 94m Change plus to minus sign of term $\int_\O \nabla u \cdot \nabla (u-w_h) {\rm d}x$
2 116m Add let $Y_h=X_h'$; replace $Y_H$ by $Y_h$
3 117b Replace $\cS^1(\cT_h)^d$ by $X_h$ (three times) in Algorithm 4.4

Chapter 5: The Obstacle Problem

Chapter 6: The Allen-Cahn Equation

Nr. Page(s) Correction
1 154t Fig. 6.1: Graph of $f$ should reflect property $f(0)=0$
2 157m The inequality $\abs{f(s_1)-f(s_2)} \le c_f \abs{s_1-s_2}$ holds for all $s_1,s_2\in [-1,1]$
3 158t Use $a$ instead of $b$ in definition $b= (1+2c_f\veps^{-2})$
4 179t Move factor $h$ in $h \norm{w-Q_h}+... $ to second term.
5 179b Factor $q_h$ is missing in last estimate of the proof (four times)

Chapter 7: Harmonic Maps

Chapter 8: Bending Problems

Nr. Page(s) Correction
1 218m Replace $H^2(\O)$ by $H^2(\omega)$ in Def. 8.2
2 224m Replace $\varphi$ by $\phi$ (four times)
3 236t The elementwise defined quantity $D^2 \tu_h - \nabla \nabla_h \tu_h$ converges to zero since $D^3 \tu_h$ is bounded, which follows from $\norm{D^3 \tu_h} \le c h^{-1} \norm{D^2(\tu_h - q_h)} \le c \norm{D^3 \tu}$ with a suitable quadratic polynomial $q_h$
4 237b Replace term $\tau F$ by $F$
5 253b Remark 8.9: The algorithm leads to families of triangulated surfaces with good mesh properties. For a related version of the algorithm for curves an equidistribution property can be proved, cf. [2].

Chapter 9: Nonconvexity and Microstructure

Chapter 10: Free Discontinuities

Nr. Page(s) Correction
1 298t Replace $v\in C_0$ by $\phi\in C_0$
2 298m Replace $\R^m$ by $\R$ in Example 10.1 (iii)
3 298b Replace $\R^n$ by $\R^d$ in Definition 10.1 (once)
4 302t Remove comma in ${\rm div}, \phi$

Chapter 11: Elastoplasticity

Nr. Page(s) Correction
1 351b The Neumann boundary term and the kinetic hardening part of the energy functional are missing in definition of $\eta_k$
2 361m The function nonlinear_fe_matrices_plast should be used in line 3 of subroutine plastic_step

Appendix A: Auxiliary Routines

Nr. Page(s) Correction
1 367 See linked file for a file generating a triangulation of a disk: triang_disk.m
2 369 See linked file for a more efficient implementation: red_refine.m

Appendix B: Frequently Used Notation