Correction to: Numerische Mathematik (2022) 150:299–371 https://doi.org/10.1007/s00211-021-01256-x

We are grateful to Andreas Rathsfeld (WIAS, Berlin) for drawing our attention to an error in the statement of one of the results of this paper. The error is in the statement of part (b) of Theorem 3.2, which claims more than is true. This has minor implications for where this result is used later in the paper, including in the statement of Theorem 3.17.

We provide, for the convenience of the reader, corrected statements of Theorems 3.2 and 3.17 at the end of this erratum. First, we provide a list of the six corrections as follows:

1. The statement of part (b) of Theorem 3.2 should read (with no initial “For some \({\textbf{x}}^*\in \Gamma \)”):

(b)

$$\begin{aligned} W_{{\textrm{ess}}}(D)= & {} \bigcap _{\delta>0} {\textrm{conv}}\left( \overline{\bigcup _{{\textbf{x}}\in \Gamma } W(D_{{\textbf{x}},\delta })}\right) = {\textrm{conv}}\left( \bigcap _{\delta>0} \overline{\bigcup _{{\textbf{x}}\in \Gamma } W(D_{{\textbf{x}},\delta })}\right) \\ {}= & {} {\textrm{conv}}\left( \bigcup _{{\textbf{x}}\in \Gamma } \bigcap _{\delta >0} \overline{W(D_{{\textbf{x}},\delta })}\right) . \end{aligned}$$

2. In the proof of part (b) of Theorem 3.2, equation (3.17) on Page 330 should be replaced by

$$\begin{aligned} S_\infty \subset \bigcup _{{\textbf{x}}\in \Gamma } W_{\textbf{x}}, \end{aligned}$$

the “Since \(W_{{\textbf{x}}^{*}}\) is convex” on the line above (3.17) should be replaced by “Since \(W_{\textrm{ess}}(D)\) is convex”, and the “for some \({\textbf{x}}^*\in \Gamma \)” on the line below (3.17) should be deleted.

3. In the statement of Theorem 3.17, in the line before equation (3.34) on page 337, the “for some \({\textbf{x}}^*\in \Gamma \)” should be deleted, and (3.34) should read:

$$\begin{aligned} W_{{\textrm{ess}}}(D) = {\textrm{conv}}\left( \overline{\bigcup _{{\textbf{x}}\in \Gamma } W(D_{{\textbf{x}}})}\right) = {\textrm{conv}}\left( \bigcup _{{\textbf{x}}\in V_N}\overline{ W(D_{{\textbf{x}}})}\right) . \end{aligned}$$

4. In the proof of Theorem 3.17, starting in the last paragraph on Page 337, the “for some \({\textbf{x}}^*\in \Gamma \)” should be deleted, the penultimate displayed equation on Page 337 should read

$$\begin{aligned} W_{{\textrm{ess}}}(D) = {\textrm{conv}}\left( \bigcup _{{\textbf{x}}\in \Gamma } \bigcap _{\delta >0} \overline{W(D_{{\textbf{x}},\delta })}\right) , \end{aligned}$$

the last displayed equation on Page 337 should read

$$\begin{aligned} W_{{\textrm{ess}}}(D) = {\textrm{conv}}\left( \bigcup _{{\textbf{x}}^*\in \Gamma } \overline{W(D_{{\textbf{x}}^*})}\right) = {\textrm{conv}}\left( \bigcup _{{\textbf{x}}^*\in \Gamma } \overline{W(D^\rho _{{\textbf{x}}^*})}\right) , \end{aligned}$$

and the last sentence of the proof, on Page 338, should read as follows: For every \({\textbf{x}}^*\in \Gamma \), \(\Gamma _{{\textbf{x}}^*}^{\rho }\subset \Gamma \cap \Gamma _{{\textbf{x}}}\), for some \(\rho >0\) and \({\textbf{x}}\in V_N\), so that, by (2.5),

$$\begin{aligned} W_{\textrm{ess}}(D)= & {} {\textrm{conv}}\left( \bigcup _{{\textbf{x}}^*\in \Gamma } \overline{W(D^\rho _{{\textbf{x}}^*})}\right) \subset {\textrm{conv}}\left( \bigcup _{{\textbf{x}}\in V_N}\overline{W(D_{{\textbf{x}}})}\right) \subset {\textrm{conv}}\left( \bigcup _{{\textbf{x}}\in \Gamma } \overline{W(D_{{\textbf{x}}})}\right) \\ {}= & {} W_{\textrm{ess}}(D). \end{aligned}$$

5. The final sentence of Section 3 on page 338 no longer makes sense and should be deleted.

6. The first displayed equation of the proof of Theorem 1.3 on page 360 should read

$$\begin{aligned} W_{{\textrm{ess}}}(D) = {\textrm{conv}}\left( \bigcup _{{\textbf{x}}\in V} \overline{W(D_{{\textbf{x}}})}\right) . \end{aligned}$$

Here are the new versions of Theorems 3.2 and 3.17, incorporating the above corrections:

FormalPara Theorem 3.2’

[Localisation result for general Lipschitz case] Suppose that \(\Omega _-\) is a bounded Lipschitz domain and let

$$\begin{aligned} D_{{\textbf{x}},\delta }:=P_\delta ({\textbf{x}})D P_\delta ({\textbf{x}}). \end{aligned}$$

(a) For some \({\textbf{x}}^*\in \Gamma \),

$$\begin{aligned} \Vert D\Vert _{L^2(\Gamma ),{\textrm{ess}}}= \lim _{\delta \rightarrow 0} \,\sup _{{\textbf{x}}\in \Gamma } \Vert D_{{\textbf{x}},\delta }\Vert _{L^2(\Gamma )} = \sup _{{\textbf{x}}\in \Gamma } \,\lim _{\delta \rightarrow 0} \Vert D_{{\textbf{x}},\delta }\Vert _{L^2(\Gamma )} = \lim _{\delta \rightarrow 0} \Vert D_{{\textbf{x}}^*,\delta }\Vert _{L^2(\Gamma )}. \end{aligned}$$

(b)

$$\begin{aligned} W_{{\textrm{ess}}}(D)= & {} \bigcap _{\delta>0} {\textrm{conv}}\left( \overline{\bigcup _{{\textbf{x}}\in \Gamma } W(D_{{\textbf{x}},\delta })}\right) = {\textrm{conv}}\left( \bigcap _{\delta>0} \overline{\bigcup _{{\textbf{x}}\in \Gamma } W(D_{{\textbf{x}},\delta })}\right) \\ {}= & {} {\textrm{conv}}\left( \bigcup _{{\textbf{x}}\in \Gamma } \bigcap _{\delta >0} \overline{W(D_{{\textbf{x}},\delta })}\right) . \end{aligned}$$
FormalPara Theorem 3.17’

[Localisation for locally-dilation-invariant surfaces and polyhedra] Suppose that \(\Gamma \), the boundary of the bounded Lipschitz domain \(\Omega _-\), is locally dilation invariant and \(V_N=\{{\textbf{x}}_1,\ldots ,{\textbf{x}}_N\}\) is a set of generalised vertices of \(\Gamma \). (This is the case, in particular, if \(\Omega _-\) is a polygon or a polyhedron and \(V_N\) is the set of vertices in the normal sense.) Then

$$\begin{aligned} \Vert D\Vert _{L^2(\Gamma ), \,{\textrm{ess}}} = \sup _{{\textbf{x}}\in \Gamma } \Vert D_{{\textbf{x}}}\Vert _{L^2(\Gamma _{{\textbf{x}}})}=\max _{{\textbf{x}}\in V_N} \Vert D_{{\textbf{x}}}\Vert _{L^2(\Gamma _{{\textbf{x}}})} \end{aligned}$$

and

$$\begin{aligned} W_{{\textrm{ess}}}(D) = {\textrm{conv}}\left( \overline{\bigcup _{{\textbf{x}}\in \Gamma } W(D_{{\textbf{x}}})}\right) = {\textrm{conv}}\left( \bigcup _{{\textbf{x}}\in V_N}\overline{ W(D_{{\textbf{x}}})}\right) . \end{aligned}$$