# Corrigenda Numerical Approximation PDEs


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
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

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

# Other corrections

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

# Acknowledgment

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