Corrigenda Numerical Approximation PDEs

$\def\R{\mathbb{R}}$ $\def\C{\mathbb{C}}$ $\def\O{\Omega}$ $\def\veps{\varepsilon}$ $\def\vphi{\varphi}$ $\def\hT{\widehat{T}}$ $\newcommand{norm}{\|#1\|}$ $\newcommand{abs}{|#1|}$

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: 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. .
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