# 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: February 7, 2019

# 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

# 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)

# 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

# Chapter 4: Local resolution techniques

Nr. Imp. Page(s) Correction
1 206 Correct author name of Reference 22: Praetorius, D.

# 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.

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 $\Omega$

# 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 $\delta_{mn}$

# 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: $... = \int_\O f\cdot v_h dx + \int_{\Gamma_N} g \cdot v_h ds - \sum ...$

# 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 $\varrho = 1.2041\, {\rm kg/m}^3$

# 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