Corrigenda Numerical Approximation PDEs

$\def\R{\mathbb{R}}$ $\def\C{\mathbb{C}}$ $\def\O{\Omega}$ $\def\veps{\varepsilon}$ $\def\vphi{\varphi}$ $\def\hT{\widehat{T}}$ $\def\Tkple{\mathcal{T}_{k+1}}$ $\newcommand{norm}[1]{\|#1\|}$ $\newcommand{abs}[1]{|#1|}$ $\newcommand{unitinterval}{[0,1]}$

The following list of corrections is updated occasionally and ordered chronologically. Important changes are marked by an exclamation mark. Letters t,m,b refer top, middle, and bottom parts of a page.

Last update: March 9, 2021

Chapter 1: Finite difference method

Nr. Imp. Page(s) Correction
1 6m Remove subscript $x$ in identities for difference quotients (3 times); replace $x$ by $x_j$ (3 times) in formula following This implies that
2 ! 14m Remove factor $\Delta x$ in denominator of identity for $\norm{V}$
3 24t Change: ... is regular with eigenvalues $\lambda_j \ge 1$, $j=1,2,\dots,J-1$, and ...
4 29t Change: ... that $f(z) \ge -1$ to deduce $\abs{f(z)} \le 1$, i.e., ...
5 37m Replace plus by minus sign in formula $3/4+\abs{x}/2$
6 ! 56m Remove minus sign in identity $Lu = -\Delta ...$
7 ! 61b Replace $A$ by $-A$ in the equation of Example 1.14
8 34-35 Improved derivation of the wave equation: wave_equation.pdf (V. 7.11.2018)
9 22t Replace U(k+1,J+1) by U(k+2,J+1) in line 17 of code explicit_euler.m
10 ! 29m Remove factor $\Delta x$ in sum.
11 32b According to the Gerschgorin theorem the eigenvalues of $\widetilde{A}$ are bounded from below by one.
12 36m Add remark: We will see that the wave speed is given by $c = (\sigma/\varrho)^{1/2}$ so that to double the frequency of a vibrating string we have to divide its weight or increase its tension by 4.
13 ! 42t,m Correct: ... if the complex eigenvalues of the matrices $A_p$ are distinct and bounded by one and Since $z_1z_2 =1$ a sufficient condition for $z_1\neq z_2$ and $\abs{z_{1,2}}\le 1$ is that $\abs{\gamma_p}<1$. Correct also: $-1 < \cos(\pi p \Delta x) < 1$ and $\abs{z_{1,2}}\le 1$
14 51 Add: If $u\in C^4(\overline{\O})$ then we have the estimate ... in Proposition 1.17
15 14m/15t Add subscript $\Delta t$ in $\abs{\widetilde{E}(\xi)}$ (three times)
16 16m Correct $\pi r^2 ... =-\kappa \partial_x u(t,x_2)-...$
17 44m Last sum should begin with summation index $j=1$
18 48t Alternatively, one may use that the integral mean of $u$ over $\partial B_r(x_0)$ equals the constant function $\varphi(r) = c_d \int_{\partial B_1(0)} u(x_0+r z) dz$
19 48m Replace cancelling factors $a^2$ by $r^2$ in formula following to obtain
20 51m Condition $u\in C^4(\overline{\Omega})$ should be added in Prop. 1.17, replace $u_{j,m}$ by $u(x_{j,m})$ and $U(x_{j,m})$ by $U_{j,m}$ in the proof
21 56m The function $G$ maps into $\R$, the identity $G(u)=Mu-\ell$ should be replaced by $G(z,u(z),...) = Mu(z) -\ell(z)$

Chapter 2: Elliptic partial differential equations

Nr. Imp. Page(s) Correction
1 67b Replace $v\in C(\overline{\O})$ by $v\in C^1(\overline{\O})$
2 81b Change: Let $\O\subset \R^d$ be bounded, $u\in L^p(\O)$, and ...
3 93m Add: If $a$ is symmetric then the factor $k_a/\alpha$ can be replaced by its square root.
4 ! 95t Remove minus sign in front of boundary integral; replace Proof by Proof (sketched)
5 ! 95m Add We incorporate the result that because of the convexity of $\O$ smooth functions are dense in the set $V$ of all functions $v \in H^1_0(\O)$ sucht that $-\Delta v\in L^2(\O)$ with respect to the norm $v \mapsto \norm{\nabla v} + \norm{\Delta v}$, cf. [5].
6 70 Add ... called a scalar or inner product ... in Definition 2.3 (i).
7 71m Add: By the parallelogram identity $2 \norm{x}^2 + 2 \norm{y}^2 = \norm{x-y}^2 + \norm{x+y}^2$ we have that ...
8 72t Replace $r$ by $z$ and use that $u-z\in U$
9 95-96 Replace $x'$ by $z'$ (five times)
10 69t Replace $u$ by $v$ in identity for $I(v)$
11 73b Add to Prop. 2.5: A bounded linear operator $\tilde{A}:\tilde{V}\to W$ defined on a dense subspace $\tilde{V} \subset V$ can be uniquely extended to a bounded linear operator on $V$.
12 80m Correct: we define the vertical average $f_k(x')= ...$ and note
13 82t For proving statement (iii), Lusin's theorem is used
14 82m Add: ... that the integration-by-parts formula ... holds for all ...
15 88m Replace $C_P$ by $c_P$ once
16 96m Replace The second equation then leads to by while the second equation simplifies to
17 96b Add to Rem. 2.17: The example $u(r,\phi) = r^{\pi/\gamma} \sin(\phi \pi/\gamma)$ from Prop. 2.1 shows that the result is sharp; $H^2$ regularity of a boundary value problem is a stronger requirement than the condition that solutions satisfy $u\in H^2(\Omega)$.

Chapter 3: Finite element method

Nr. Imp. Page(s) Correction
1 ! 106b Replace factor $h_T^{-k}$ by $h_T^k$ in Proposition 3.3
2 108m Item (v): Replace first integration domain by $T$ and second by $\hT$ in first formula and first integration domain by $\hT$ in second formula
3 108b Change $\chi$ to $\widehat{\chi}$
4 ! 111t Replace $C^{m-1}$ by $C^r$ and correct $0\le k\le \min\mbox{{$r+1,m$}}$
5 116b Change $A^{-1} = (I-M)^{-1} D^{-1}$
6 118m Add: Assume $u_h\in V_h$ is such that $a_h(u_h,v_h) = \ell(v_h)$ for all $v_h\in V_h$.
7 131 Replace variables (nS,idx_Bdy,newCoord) by (nEdges,idxBdy,newCoords)
8 151m Use command "m_lumped = spdiags(m_lumped_diag,0,nC,nC);"
9 103b Add: By integrating the gradient field $w=(w_1,w_2,\dots,w_d)$ along appropriate paths ...
10 117 Add: , i.e., the minimum is attained on the boundary. at end of Proposition 3.5
11 118 Replace $c>0$ by $c_{SL}>0$ in Proposition 3.6
12 ! 118b The estimate for the $\abs{\ell(v_h)-\ell(v_h)}$ is suboptimal; a better result is provided in the pdf file quadrature.pdf
13 ! 122b Remove squares of first two norms
14 ! 144 Add prefactor $\tau$ to the sum in Proposition 3.13, 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 $\partial_t^2 u$, respectively, in the proof
15 140,145 The solutions should be sought on the closed time interval $[0,T]$ in Definitions 3.13 and 3.15
16 107b Add subscript $T$ in $\Phi$ twice
17 109m Correct $\widehat{v} = v \circ \Phi_T \in W^{m,p}(\widehat{T})$ and $\abs{\widehat{v}}_{W^{m,p}(\widehat{T})}$
18 106t Replace $R$ by $R+1$ in estimate for $\abs{F(v)}$
19 111t The estimate follows from Proposition 3.3 by summing over all elements $T\in \mathcal{T}_h$.
20 99b A line break should be introduced before The integer ...
21 100t The functional $\chi_T$ should be placed in parantheses
22 103t Remove power $p$ in $W^{m,p}$ norm of $v$
23 107b Remove last equal term in equation for $\nabla w$
24 108t Include strict lower bound for $\varrho_{\hat{T}}$
25 ! 108m Replace $T$ by $\hat{T}$ in final upper bound of (v), in addition to Nr. 2 above
26 109t Replace $P_T$ by $\mathcal{P}_T$
27 121b Replace $f$ by $\tilde{f}$ in discrete weak formulation
28 ! 122b Replace integration domains $\Omega \setminus \Omega_h$ by $\Omega_h \setminus \Omega$ three times
29 ! 124m Correct: Otherwise, if $p>r$ use the monotonicity property of the $\ell^p$ norm $\norm{x}_p \le \norm{x}_r$ for every ...
30 126b Correct $\Gamma_D = \unitinterval \times {0}$ in Fig. 3.10
31 136b Equivalently, we have $v_h \circ \psi_T \in \mathcal{P}_2(T)$ for all $T\in \mathcal{T}_h$
32 137t The second equality sign should be replaced by the approximation symbol $\approx$
33 144m The first displayed formula in the proof should be replaced by the representation $(d_t Q_h u(t_k)-\partial_t u(t_k),v_h)$ of the consistency term
34 147m In the estimate of Prop. 3.16 the maximum over $k=1,...,K$ should be considered

Chapter 4: Local resolution techniques

Nr. Imp. Page(s) Correction
1 206 Correct author name of Reference 22: Praetorius, D.
2 155m Add: solution of the Poisson problem with vanishing right-hand side
3 159m Correct: is positive if $\alpha \neq 0$
4 ! 162b The patch $\omega_z$ should be the interior of the support, i.e., $\omega_z = {\rm int}\,{\rm supp} \, \vphi_z$
5 169m Add in Def. 4.6: $f\in L^2(\O)$
6 ! 172b Correct in Lemma 4.3: ${\rm supp}\, b_T = T$ and ${\rm supp}\, b_S = \overline{\omega}_S$
7 182 Throughout Section 4.3 the index $k$ belongs to $\mathbb{N}_0$
8 ! 182b Correct: if every $T\in \mathcal{T}_k$ is the union of elements in $\Tkple$
9 188m Note that $\delta_{k+1}$ vanishes on $\partial \O$; for the definition of the Scott-Zhang interpolant on $\O_i$ we assume that sides $S_z$ associated with $z\in \partial \O_i$ also belong to $\partial \O_i$
10 ! 189b Change $2^{-1/2}$ to $2^{-1/d}$ and replace $\sqrt{2}$ and $2^{1/2}$ by $2^{1/d}$ throughout Subsection 4.3.4
11 192t The sum is over elements $T'\in \mathcal{T}_{k+1}$ with $T'\subset \overline{\O \setminus \cup \mathcal{M}_k}$
12 ! 193b Replace exponent $-d/s$ by $-d/(2s)$
13 194m Add factor $c_{{\rm card}}$ after second inequality and replace $c_{qo}$ by $\widetilde{c}_{qo}$
14 196b Use $\veps = \widetilde{q} \norm{\nabla (u-u_k)}$ and note that $e_k \approx \norm{\nabla(u-u_k)}$
15 201m Add in initialization of Alg. 4.4: set $\mathcal{T}^1= \mathcal{T}^0$ and $j=1$; the stopping criterion in Step (3) should require that $\eta_{{\rm temp}}^j \le \veps$; the otherwise case in Step 4 should set $\mathcal{T}^j = \mathcal{T}^{j-1}$
16 204t Replace the abstract theorem by Proposition 4.5 and bounds for $u-\widetilde{u}$ by $\widetilde{u}-U$; add noting that $-\Delta \partial_t \widetilde{u} = -\Delta_h \partial_t U$
17 205m Replace the abstract error estimate by Proposition 4.5
18 ! 169t The convergence $\mathcal{J}_h v \to v$ is in $L^2(\Omega)$
19 170/171 Replace square brackets $[..]$ by double square brackets denoting jumps three times
20 172t Correct: *The jumps of $\nabla u_h$ are a measure of nonsmoothness of $u_h$ *
21 174t Replace square brackets $[..]$ by double square brackets denoting jumps twice
22 178m Here, $\varrho_z$ is the height of the element with respect to the side $S_z$
23 ! 179b Add square to factor $f(..)$ in formula for $\eta_T$; remove factor $h_T^{(2-d)/2}$ in formula for $\eta_S$
24 186m/b Replace $h_{S_z}$ by $h_z$ and remove second power $\ell$ in term $\nabla^\ell \varphi_z^\ell$
25 187t Exchange order of last two terms in right-hand side of first bound
26 188m Remove fullstop in front of and note
27 ! 189t Remove minus sign in front of last sum
28 191t Replace square brackets $[..]$ by double square brackets denoting jumps once

Chapter 5: Iterative solution methods

Nr. Imp. Page(s) Correction
1 210 Add in Remark 5.1: This follows from the explicit characterization of eigenfunctions of the discretized Laplace operator on uniform grids, cf. Lemma 1.1 in Chapter 1.
2 219m Correct sign in exponent: $2^{\ell d} = n_L 2^{(\ell-L)d}$, analogously in sum below
3 224b Replace $v\in H^1_0(\O_1)$ by $v\in H^1_{\Gamma_1}(\O_1)$
4 227t Add lower bound: $0<\theta<\theta* = ...$
5 230m Add lower bound: $\norm{\nabla v} \le \norm{\nabla v_1}_{L^2(\O_1)} +\, ... $
6 235m A simpler derivation considers the eigenvalues of $C^{1/2}A C^{T/2}$ via a standard (euclidean) Rayleigh quotient and the transformation $v=C^{T/2} w$
7 237t Note that the right-hand side of the estimate is bounded by $c h \norm{\nabla v_h}$
8 243t The case that $\mathcal{T}_H$ has no free nodes formally corresponds to an infinite mesh size $H = \infty$
9 216t Correct Computing

Chapter 6: Saddle-point problems

Nr. Imp. Page(s) Correction
1 251m Add in Remark 6.2 (ii): ... $ \norm{Mx} \ge \gamma \norm{x}_\ell$ for all $x\in \R^n$, i.e., $M$ is injective, hence regular, and to ...
2 253m Add: and analogously, noting $B_K=0$,... and $B^Ts = [0,B_I^Ts]^T\equiv B_I^Ts \in \text{Im} B^T$
3 ! 253b Exchange first and second conditions and enumerate them.
4 265b Add in Theorem 6.4: ... if and only if $a$ is coercive on $\text{ker}B= \text{{$v\in V: b(v,q) = 0 \text{ f.a. }q\in Q$}}$
5 267 Correct: ... let $u\in \text{ker}B \setminus \text{{0}}$; change With $p=0$ we have that $(u,p)=L^{-1}(\ell,0), i.e., ...$ correct $\sup_{v\in V\setminus \text{{0}}}$ after show that
6 274m Change: In fact, a simpler error estimate that resembles the classical Cea lemma can be proved if ...
7 ! 274b Insert condition ${q_h}\rvert_{\partial \Omega} = 0$ in definition of $Q_h$
8 ! 276m Delete such that there exist $v\in V$ and $q_h\in Q_h$ with $b(v,q_h)\neq0$ and and nontrivial
9 279t Add remark: Note that $v_h \mapsto \norm{Bv_h}_Q$ defines a norm on $V_h$ so that $\norm{B v_h}_Q \le c_h \norm{v_h}$, where $c_h$ depends on the dimension of $V_h$
10 279m Add: ... we expect $\sigma = 1$ since $\norm{\text{div} I_h v} \le c h \norm{D^2 v}$ for every smooth divergence-free vector field $v$ on $\Omega$
11 273t Add subindex $X$ to norm $\norm{x-w_h}$
12 274t The part of Theorem 6.7 beginning with Then for every ... should not be indented
13 276m The subspaces $V_h$ and $Q_h$ should be closed

Chapter 7: Mixed and nonstandard methods

Nr. Imp. Page(s) Correction
1 ! 317 Corrections in lines 21, 36, 45 of code "stokes_cr", see file stokes_cr_corr.m
2 284m Correct Remarks 7.1 (ii): The set $C^\infty(\overline{\Omega};\R^d)$ is dense in $H(\text{div};\Omega)$.
3 286b Replace factor $(1+c_P^2)$ by $(1+c_P^2)^{1/2}$ twice
4 288b Insert: But then $u_h(x) \cdot n_{S_{d+1}} = d_T (x-z_{d+1}) \cdot n_{S_{d+1}} = 0$ for every ...
5 289m Replace $x\in T_+\cup T_-$ by $x\in \Omega$
6 290m Add: ... has to be used which maps normals to normals.
7 292m Replace ... allows us to prove ... by ... allows us to deduce ...
8 293m Add comment: The smallness condition on $h$ cannot be avoided in general; for $h$ sufficiently small we have that the number of sides dominates the number of elements in $\mathcal{T}_h$.
9 293m Replace $v_h$ by $-v_h$ in Rem. 7.6
10 294t Replace $H^1(T)$ by $H^1(T;\R^d)$ twice
11 295b Add in Rem. 7.2 (ii): ... provided that the additional regularity condition $g\in H^1(\Omega)$ is satisfied.
12 305t Proof of the discrete inf-sup condition for the Taylor-Hood element: taylor_hood.pdf
13 305m Lemma 7.3: Add There exists $\beta'>0$ such that for all $q_h \in Q_h$ we have ...; define $w_z = 0$ for $z\in \Gamma_D = \partial \Omega$
14 307b Delete unnecessary brackets $(\beta'...)$ and incorrect term $-\mu \norm{\nabla u_h} \norm{p_h}_{L^2(\Omega)}$
15 309t Change second sentence in Rem. 7.10 (ii): Here, we use a special bilinear form $c_h$ and are able to avoid an inf-sup condition for $b$.
16 311b Write $b_h$ instead of $b$ for discretized bilinear form
17 312b Correct: Since $I_F$ reproduces constants we may replace $v$ by $v-\overline{v}_T$ ...
18 324m Insert: ... and in case that $\Gamma_N$ is empty in fact ...
19 328b Correct: Applying Gauss's theorem on inner sets $\Omega_j$ or imposing $q(u)\cdot n = 0$ on $\partial \O \cap \partial \O_j$ yields that ...
20 334m Insert dots for scalar products with vector $n_S$ twice
21 343b Move factor $d$ inside brackets in front of Kronecker symbol $\delta_{mn}$
22 ! 289m Replace factor $d!$ by $d$ in formula for $\delta_{S,T}$
23 316b The loop over $k$ is misleading and should be removed, replace factor $1/d$ by $1$

Chapter 8: Applications

Nr. Imp. Page(s) Correction
1 403 Add reference M. Costabel, A coercive bilinear form for Maxwell’s equations, Journal of Mathematical Analysis and Applications, 157 (1991), 527–541.
2 349m Correct: ... two nearby material points ...
3 354t Modify: then we obtain robust approximation results in the relative error for the limit $\lambda \to \infty$
4 356m Add in Rem. 8.4: which is equivalent to $\int_\O \mathbb{C} \veps_{\mathcal{T}} (u_h) : \veps_{\mathcal{T}} (v_h) dx = \ell(v_h)$
5 358t Correct: $a_h(u,v_h) = \sum_{T\in \mathcal{T}_h} \int_T \mathbb{C} \veps(u): \nabla v_h d x$
6 358t Correct two signs: $... = \int_\O f\cdot v_h dx + \int_{\Gamma_N} g \cdot v_h ds - \sum ...$
7 356m The condition $\lambda \ge 1$ should be added to Prop. 8.2
8 357m The triple bar expression is a norm provided $\Gamma_D$ contains at least two sides $S\in \mathcal{S}_h$
9 357b It is assumed that $\beta_S \ge 2\mu$ in Lemma 8.3
10 ! 358t All minus signs in final right-hand side of the identities for $a_h(u_h,v_h)$ should be changed to plus signs (see also Nr. 6)
11 359t Sides belonging to $\overline{\Gamma}_N$ should not be included in the last sum
12 359m Remove factor $\beta_S$ in front of $h_T^2$
13 ! 362t Replace factor $d!$ by $d$ in formula for $vol_S$
14 363t/m Add in captions of Fig. 8.5 and Fig. 8.6 that solutions of the Navier-Lame equations are approximated

Appendix A: Problems and projects

Nr. Imp. Page(s) Correction
1 ! 406 A.1.6: add factor $1/2$ to right-hand side
2 ! 407 A.1.8: correct $\phi_\ell(x) = e^{i\ell x}$
3 ! 410 A.1.21: replace $a_n$ and $b_n$ by $\alpha_n$ and $\beta_n$
4 ! 412 A.1.28: correct right-hand side $\frac12 \partial_t^-(\partial_t^+ U_j^k)^2$
5 ! 422 A.2.1: replace term $r^{-1} \widetilde{u}(r,\phi)$ by $r^{-1} \partial_r \widetilde{u}(r,\phi)$ in formula for $\Delta u(r,\phi)$
6 ! 429 A.2.33: remove factors $1/N$ (twice)
7 ! 438 A.3.26: where $\varrho_j>0$ is the inverse of the height of $T$ ...
8 ! 407 A.1.8.: Assume that $f=\sum_{\ell\in \mathbb{Z}} f_\ell \phi_\ell$ in second part of (ii)
9 411 A.1.26: Hint: Use that the spectral radius is a sharp lower bound for operator norms
10 ! 420 A.1.5: Correct $c = (\sigma/\varrho)^{1/2}$ and use the cosine transform to compute the coefficients $\alpha_m$
11 ! 421 A.1.8: Correct density $\varrho = 1.2041\, {\rm kg/m^3}$
12 ! 443 A.3.8: Correct density $\varrho = 1.009\cdot 10^3 \, {\rm kg/m^3}$
13 ! 444 A.4.2: Move minus sign from second entry of second row to first entry of second row in matrix
14 450 A.4.32: Add: for sufficiently regular weak solutions
15 450 A.4.33: Add $\frac12 \norm{\nabla U^0}^2$ to right-hand side in part (ii)
16 457 A.5.12: Correct: $1/2< a< 1$

Appendix B: Implementation aspects

Nr. Imp. Page(s) Correction
1 509 Add command "d_fac = factorial(d)" in function "nodal_basis" and correct identity for "Vol_T(j)" to improve performance

Appendix C: Notation, inequalities, guidelines

Other corrections

Nr. Imp. Page(s) Correction
1 iv Correct Universität


Thanks for valuable hints to: D. Gallistl, L. Minden, C. Palus, G. Dolzmann