Corrigenda Numerical Methods Nonlinear PDEs

$\def\R{\mathbb{R}}$ $\def\C{\mathbb{C}}$ $\def\N{\mathbb{N}}$ $\def\O{\Omega}$ $\def\o{\omega}$ $\def\cS{\mathcal{S}}$ $\def\cT{\mathcal{T}}$ $\def\veps{\varepsilon}$ $\def\vphi{\varphi}$ $\def\hT{\widehat{T}}$ $\def\tu{\widetilde{u}}$ $\newcommand{norm}[1]{\|#1\|}$ $\newcommand{abs}[1]{|#1|}$ $\def\b{\beta}$ $\def\d{\delta}$ $\def\hu{\widehat{u}}$ $\def\GD{\Gamma_{\rm D}}$ $\def\GN{\Gamma_{\rm N}}$ $\def\uhs{u_h^*}$ $\def\diver{\rm div}$ $\def\wto{\rightharpoonup}$ $\def\OminAd{\O\setminus A_\d}$

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: March 17, 2022

Chapter 1: Introduction

Nr. Page(s) Correction
1 7b Remove redundant and in image processing and and fracture

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
3 20m Remove redundant comma before equality sign in first displayed formula
4 21m Replace $C^\infty_0(\O)$ by $C^\infty_c(\O)$
5 22t Replace (first) inequality $p>d$ in line 2 by $p<d$
6 22m Add item: (ix) Egorov's theorem states that if $u_n \to u$ pointwise almost everywhere then $u_n \to u$ almost uniformly, i.e., for every $\d >0$ there exists $A_\d \subset \O$ with $\abs{A_\d}\le \d$ such that $\norm{u_n-u}_{L^\infty(\OminAd)} \to 0$. This implies that if $u_n \to u$ pointwise almost everywhere and $(u_n)_n$ is bounded in $L^p(\O)$ for $1<p<\infty$ then $u_n\wto u$ in $L^p(\O)$
7 23t Replace $a,b\in \R$ by $a,b\ge 0$
8 24m Remark 2.4 (ii): The function $j$ has to be a Caratheodory function and one needs that $1\le q < p^*$, a constant $c$ is allowed in front of the term $\abs{s}^q$ in the estimate for $\abs{j(x,s)}$
9 33t Replace $X$ by $X'$ in $v'\in L^{p'}([0,T];X)$
10 33m Replace $L^2(I;H)$ by $L^2([0,T];H)$ in Remarks 2.15 (v)
11 35m Replace $\ell$ by $L$ in last formula of Lemma 2.2
12 36b Remove factors $1/2$ after both inequality signs in formula following This shows that; remove third factor $c_P^p$ in last displayed formula
13 38m Replace $u_{\tau_n}$ by $u_{\tau_n}^\pm$ twice in Proposition 2.3
14 40t The convergence holds for $f(\hu_{\tau_n})$, additional arguments are required to establish this with $u_{\tau_n}^+$
15 22m The trace operator maps into $L^p(\partial \O,\R^m)$ in Remarks 2.3 (iii)
16 22b Correct $D^ku = (\partial^\alpha u)_{\abs{\alpha} = k}$
17 25t Remove subscript from $\widetilde{\chi}_{(\theta,1)}$ in integral
18 27m/b Replace $W$ by $W_0$ once, $r$ by $t$ in second integral representation for $M^r$, and $t\in (0,r)$ by $r\in (0,1)$
19 32t/b Add subscript $X$ to norms $\norm{u(t)}$ twice
20 38t Replace $u^+$ by $u_\tau^+$ in first estimate of proof
21 42m Replace $I$ by $I_{\lambda,w}$ in it follows that $I$ is weakly lower semicontinuous

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)
5 71b Add prefactor $\tau$ to the sum in Proposition 3.8; replace factors $ch^2$ and $c\tau$ by $c\tau^{-1/2}h^2$ and $\tau^{1/2}$, respectively, and terms $u_t$ and $u_{tt}$ by $\partial_t u$ and $\partial_t^2u$, respectively, in the proof

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
4 90m The convergence follows from a local Lipschitz continuity estimate $\abs{W(A)-W(B)} \le c_W (1+\abs{A}^{p-1}+\abs{B}^{p-1}) \abs{A-B}$ assuming that $W$ is $C^1$ and satisfies a $(p-1)$-growth condition
5 91b The convergence $I_h(u_h)\to I(u)$ follows directly from the inequality $\abs{(h^\b+\abs{\nabla u_h}^2)^{1/2}- \abs{\nabla u}}\le h^{\b/2} + \abs{\nabla u_h- \nabla u}$ and $u_h \to u$ in $W^{1,1}(\O)$
6 117t Replace $DF$ by $DF_h$ and correct closing norm symbol $\norm{...}_{Y_h}$
7 117t Replace $v/\abs{v}$ by $v/\norm{v}_X$ in Remarks 4.21 (ii)
8 117b Replace $G$ by $G_h$ in second line of Theorem 4.11
9 118t Correct sentence: If $\norm{w_h^0}_X \le \veps$ with $\veps >0$ such that $\vphi(s) \le s/(2M)$ for all $0\le s\le \veps$ then we inductively ...; replace $\phi(\norm{w_h^j}_X)$ by $\vphi(\norm{w_h^j}_X)$

Chapter 5: The Obstacle Problem

Nr. Page(s) Correction
1 131b Add assumption in Proposition 5.1: Assume that the contact zone $\mathcal{C}$ is the finite union of closed intervals.
2 134m Replace condition $\xi \ge \partial_n \chi$ by $\partial_n \chi \le 0$ on $\GN$
3 139m The function $v_h$ constructed in the proof of Proposition 5.2 does in general not vanish on $\GD$; it is convenient to assume that smooth functions are dense $K$, that $\norm{\chi_h - \chi}_{L^\infty(\O)} \to 0$, and that $\chi$ is strictly negative on $\GD$; a coupling of $\veps$ and $h$ occurs in the proof
4 139b Assume that $\chi\le \chi_h$ on $\GD$ in Theorem 5.8
5 144m Proof of Lemma 5.3, correct signs in second case: ... then $0=...= \Lambda_i - \Lambda_i - c(U_i-Z_i) = -c(U_i-Z_i)$, i.e., $U_i=Z_i$ and $\Lambda_i< 0$
6 144b Proof of Theorem 5.11: The inverse of the regular matrix $DF(U,\Lambda)$ is a bounded linear operator on the finite dimensional space $\R^L\times \R^L$ whose operator norm depends on the dimension $L$.

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)
6 155/156 Replace $(\partial_t \d,v)$ by $\langle \partial_t \d,v\rangle$ (twice)
7 160m Add factors $\veps^{-2}$ in front of $f(u(t))$ and $f'(u(t))v$
8 160m Replace $c_P^2$ by $c_P^{-2}$ and $1+\veps^{-2}$ by $1+c_f \veps^{-2}$ in Remarks 6.4 (ii)
9 161t Correct sign $-\langle \widetilde{\mathcal{R}},\d\rangle$ in second displayed formula
10 165t Replace $v_t$ by $\partial_t v$
11 170m Replace $u^L$ and $u^0$ by $U^L$ and $U^0$ in estimate of Proposition 6.4
12 172b Remove factor $\veps^2$ in last displayed formula
13 173m Replace equality by inequality sign, factor $(1-\veps)$ by $(1-\veps^2)$, and $\norm{e^k}^2$ by $2 \norm{e^k}^2$ in formula following ... imply that; replace factor 48 by 24 in formula (6.2) and following identity for $y_3^k$
14 174t Add: Noting that owing to the conditon on $\tau$ we have that $\tau \mu_\lambda^k \le 1/2$, that $e^0=0$, and ... before formula (6.3)
15 175b Replace $u^k$ by $u^{k-1}$ in formulas for $F(u^{k-1})$ and $F^{cx}(u^{k-1})$
16 176b Replace factors $7/2$ and $2/7$ by $13/2$ and $2/13$ in Proposition 6.6
17 180t Replace $W_0$ by $W^0$ three times
18 181m The routine vector_iteration should be called with the matrix m_lumped
19 179m Add minus sign to scalar product $(Q_hw,Q_hw-w)$ in second equation following we find that
20 167m Add minus sign to $\Delta^{-1} w$
21 155m The maximum principle follows from testing the equation with $v_+= \max(u-1,0) $ and $v_-= \min(u+1,0)$ and noting, e.g., $2 (\partial_t u,v_+) = \frac{d}{dt} \norm{(u-1)_+}^2$

Chapter 7: Harmonic Maps

Nr. Page(s) Correction
1 188b The function $u$ in Theorem 7.4 also satisfies $u(t,\cdot)= u_{\rm D}$ on $\GD$ for almost every $t\in [0,T]$
2 191m Replace $w_h^0$ and $w_h^r$ by $u_h^0$ and $u_h^r$, respectively
3 192t Replace $\R^m$ by $\R^3$ in Theorem 7.6
4 192m Replace term $w_h - u_h \times w$ by $w_h - u_h \times \phi$
5 194t Replace factor $(1+\theta \tau)$ by $(1+\theta \tau + c \tau^2 h_{min}^{-1})$
6 194b The proof of the residual estimate does not consider the projection step; for the difference between the nodewise projected function $P_h u_h$ and $u_h = \uhs$ we have $\norm{\nabla (P_h u_h - u_h)} \le c h_{min}^{-1}\tau^2 \norm{\nabla v_h}^2$ which follows from an inverse estimate, the bound $\abs{P_hu_h(z)-u_h(z)}\le (\tau^2/2) \abs{v_h(z)}^2$, and the Sobolev inequality $\norm{v_h}_{L^4(\O)} \le c \norm{\nabla v_h}$

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].
6 227m Replace $H^3(\O)$ by $H^3(\o)$
7 236b Remove factor $\alpha$ in Algorithm 8.1

Chapter 9: Nonconvexity and Microstructure

Nr. Page(s) Correction
1 278t The convergence $I^{qc}(v_h) \to I^{qc}(v)$ follows from a local Lipschitz estimate $\abs{W^{qc}(A)-W^{qc}(B)} \le c_{W^{qc}}(1+\abs{A}^{p-1}+\abs{B}^{p-1}) \abs{A-B}$ assuming that $W^{qc}$ is $C^1$ and satisifes a $(p-1)$-growth condition

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 ${\rm div}, \phi$
5 301b Add condition that $\O_1$ and $\O_2$ are Lipschitz domains in Proposition 10.1
6 306t Strong duality can be proved by showing that given a solution $p$ of the dual problem we have that $u=g+(1/\alpha) \diver p$ solves the primal problem
7 307t Replace $\chi_{B_r}(0)$ by $\chi_{B_r(0)}$
8 308m Second norm in displayed formula should by $L^1$ norm
9 315t Replace first equality sign by inequality in formula following we deduce that
10 299b Replace first integration domain $(0,1)$ by $(-1,0)$ in equation for $\langle Du, \phi \rangle$
11 303m Add subscript $n$ to $u$ in *... and $u(x)=1$ for ... *
12 309b Include factor $\abs{Du}(\O)(1+\norm{u}_{L^\infty(\O)})$ in upper bound of Theorem 10.7
13 310m Add factor $c$ to upper bound $h^{1/2}$ in Remarks 10.10 (i)
14 315b Replace term $-\tau \theta^2$ by $-\tau^2 \theta^2$
15 320b Proposition 10.9: add assumption $\O$ star-shaped and $g\in L^\infty(\O)$ and include factor $(1+\abs{Du}(\O)(1+\norm{u}_{L^\infty(\O)}))$ in upper bound
16 324b Replace factor $c_0^{1/2}$ by $c_0$ in last estimate of proof
17 325b Replace set $S_{k-1}+1/2$ by $S_{k-1}+1/2^k$ in identity for $S_k$
18 302b The stated dependence on the diameter in the Poincare inequality only holds if $p=1$

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


Thanks for valuable hints to: M. Schedensack, L. Minden, H. Garcke