On optimal $L^2$- and surface flux convergence in FEM

Abstract

We show that optimal $L^2$-convergence in the finite element method on quasi-uniform meshes can be achieved if, for some $s_0 > 1/2$, the boundary value problem has the mapping property $H^{-1+s} \rightarrow H^{1+s}$ for $s \in [0,s_0]$. The lack of full elliptic regularity in the dual problem has to be compensated by additional regularity of the exact solution. Furthermore, we analyze for a Dirichlet problem the approximation of the normal derivative on the boundary without convexity assumption on the domain. We show that (up to logarithmic factors) the optimal rate is obtained.

Coverings

In this appendix, the distance $\text {dist}(x,M)$ for some set $M$ appears frequently. For notational convenience, we set $\text {dist}(x,\emptyset ) = 1$ to include the degenerate case $M = \emptyset$.

We quote from [15, Lemma A.1]:

Lemma A.1

Let ${\varOmega }\subset {\mathbb {R}}^d$ be bounded open and $M=\overline{M}$ be a closed set. Fix $c \in (0,1)$ and $\varepsilon \in (0,1)$ such that

$$\begin{aligned} 1 - c(1+ \varepsilon ) =: c_0 > 0. \end{aligned}$$

(A.1)

For each $x \in {\varOmega }$, let $B_x:= \overline{B}_{c \, \mathrm{dist}(x,M)}(x)$ be the closed ball of radius $c \text { dist}(x,M)$ centered at $x$, and let $\widehat{B}_x:= \overline{B}_{(1+\varepsilon ) c \, \mathrm{dist}(x,M)}(x)$ denote the stretched (closed) ball of radius $(1+\varepsilon )c \text { dist}(x,M)$ also centered at $x$.

Then there exists a countable set $x_i \in {\varOmega }$, $i\in {\mathbb {N}}$, and a constant $N \in {\mathbb {N}}$ depending solely on the spatial dimension $d$ with the following properties:

• 1. (covering property) $\cup _{i \in {\mathbb {N}}} B_{x_i} \supset {\varOmega }$;

• 2. (finite overlap on ${\varOmega }$) for each $x \in {\varOmega }$, there holds $\text {card} \{i\,|\, x \in \widehat{B}_{x_i} \} \le N$.

Proof

[15, Lemma A.1] assumed that $M \subset \overline{{\varOmega }}$. However, an inspection of the proof shows that this is not necessary. $\square$

Before we proceed with variants of the covering result of Lemma A.1, we introduce the notation of sectorial neighborhoods relative a singular set $M$:

Definition A.2

(sectorial neighborhood) Let $e$, $M \subset {\mathbb {R}}^d$ and $\widetilde{c} > 0$. Then

$$\begin{aligned} S_{e,M,\widetilde{c}}:= \cup _{x \in e} B_{\widetilde{c} \, \mathrm{dist}(x,M)}(x) \end{aligned}$$

is a sectorial neighborhood of the set e relative to the singular set M.

We are interested in coverings of lower-dimensional manifolds by balls whose centers are located on these manifolds:

Lemma A.3

Let $d \in {\mathbb {N}}$ and $1 \le d^\prime < d$. Let $\omega \subset {\mathbb {R}}^{d^\prime }$ and let ${\varOmega } \subset {\mathbb {R}}^d$ be the canonical embedding of $\omega$ into ${\mathbb {R}}^d$, i.e., ${\varOmega }:= \omega \times \{0\} \times \cdots \times \{0\} \subset {\mathbb {R}}^d$. Assume the hypotheses and notation of Lemma A.1. Then there are $\widetilde{c}> 0,\, N > 0$, and a collection of balls $B_{x_i},\, i \in {\mathbb {N}}$, as described in Lemma A.1 such that

• (i) (covering property for ${\varOmega }$) $\cup _{i \in {\mathbb {N}}} B_{x_i} \supset {\varOmega }$.

• (ii) (covering property for a sectorial neighborhood of ${\varOmega }$) $\cup _{i \in {\mathbb {N}}} B_{x_i} \supset S_{{\varOmega },M,\widetilde{c}}$.

• (iii) (finite overlap property on ${\mathbb {R}}^d$) for each $x \in {\mathbb {R}}^d$, there holds $\text {card} \{i\,|\, x \in \widehat{B}_{x_i} \} \le N$.

Proof

We employ the result of Lemma A.1 for the lower-dimensional manifold $\omega$ noting that $B_{x} \cap \omega$ is a ball in ${\mathbb {R}}^{d^\prime }$. In order to be able to ensure the covering condition for the sectorial neighborhood of ${\varOmega }$ stated in (iii), we introduce the auxiliary balls $B^\prime _{x}:= \overline{B}_{c/2 \, \mathrm{dist}(x,M)}(x)$ of half the radius. Applying Lemma A.1 with these balls $B^\prime _{x}$ and the stretched balls $\widehat{B}_x$ therefore produces a collection of centers $x_i \in {\varOmega }$, $i \in {\mathbb {N}}$, such that

• 1. $B^\prime _{x_i} \cap {\varOmega }$ covers ${\varOmega }$;

• 2. for the stretched balls $\widehat{B}_{x_i}$, we have a finite overlap property on ${\varOmega }$:

  $$\begin{aligned} \forall x \in {\varOmega }: \quad \text {card}\{i\,|\, x \in \widehat{B}_{x_i} \} \le N. \end{aligned}$$

  (A.2)

We next see that the balls $\widehat{B}_{x_i}$ even have the following, stronger finite overlap property:

$$\begin{aligned} \forall x \in {\mathbb {R}}^d: \quad \text {card}\{i\,|\, x \in \widehat{B}_{x_i} \} \le N. \end{aligned}$$

(A.3)

To see this, define the infinite cylinders $\widehat{C}_{x_i}:= \{x\,|\, \pi _{d^\prime }(x) \in \widehat{B}_{x_i}\cap {\varOmega }\}$, where $\pi _{d^\prime }$ is the canonical projection onto the hyperplane $\{x = (x_1,\ldots ,x_d) \in {\mathbb {R}}^d\,|\, x_{d^\prime +1} = \cdots = x_d = 0\}$. Clearly, $\widehat{B}_{x_i} \subset \widehat{C}_{x_i}$. These infinite cylinders have a finite overlap property by (A.2) as can be seen by writing any $x \in {\mathbb {R}}^d$ in the form $x = (\pi _{d^\prime }(x),x^{\prime })$ for some $x^\prime \in {\mathbb {R}}^{d - d^\prime }$ and then noting that $x \in \widehat{C}_{x_i}$ implies $\pi _{d^\prime }(x) \in \widehat{B}_{x_i} \cap {\varOmega }$.

Is remains to see that the balls $B_{x_i}$ cover a sectorial neighborhood of ${\varOmega }$. To that end, we note that the balls $B^\prime _{x_i}$ cover ${\varOmega }$. Furthermore, for each $x \in {\varOmega }$, we pick $x_i$ such that $x \in B_{x_i}^\prime \subset B_{x_i}$. Since the radius of $B_{x_i}$ is twice that of $B^\prime _{x_i}$, we even have $B_{c/2 \, \mathrm{dist}(x_i,M)}(x) \subset B_{x_i}$. Furthermore, by $c \in (0,1)$, we have $0 < (1 - c/2)\text { dist}(x_i,M) \le \text { dist}(x,M) \le (1 + c/2) \text { dist}(x_i,M)$. Therefore, there is $\widetilde{c} > 0$ such that $B_{\widetilde{c} \, \mathrm{dist}(x,M)}(x) \subset B_{x_i}$ and thus

$$\begin{aligned} \cup _{x \in {\varOmega }} B_{\widetilde{c} \, \mathrm{dist}(x,M)}(x) \subset \cup _i B_{x_i}. \end{aligned}$$

$\square$

We next show covering theorems for polygons and polyhedra. In the interest of clarity of presentation, we formulate two separate results. Before doing so, we point out that balls with center located on the boundary of the polygon/polyhedron ${\varOmega }$ will feature importantly so that the intersection of this ball with ${\varOmega }$ will be of interest. We therefore introduce the following notions:

Definition A.4

(solid angles and dihedral angles)

• 1. Let ${\varOmega } \subset {\mathbb {R}}^2$ be a Lipschitz polygon. Let $A$ be a vertex where the edges $e_1,\, e_2$ meet. We say that the set $B_{\varepsilon }(A) \cap {\varOmega }$ is a solid angle, if $\partial (B_{\varepsilon }(A) \cap {\varOmega }) \cap \partial {\varOmega }$ is contained in $\{A\} \cup e_1 \cup e_2$.

• 2. Let ${\varOmega } \subset {\mathbb {R}}^3$ be a Lipschitz polyhedron. Let $A$ be a vertex of ${\varOmega }$. We say that the set $B_\varepsilon (A) \cap {\varOmega }$ is a solid angle, if $\partial (B_\varepsilon (A) \cap {\varOmega }) \cap \partial {\varOmega }$ is contained in the union of $\{A\}$ and the edges and faces meeting at $A$.

• 3. Let ${\varOmega } \subset {\mathbb {R}}^3$ be a Lipschitz polyhedron. Let $e$ be an edge of ${\varOmega }$, which is shared by the faces $f_1,\, f_2$. Let $x \in e$. We say that the set $B_\varepsilon (x) \cap {\varOmega }$ is a dihedral angle, if $\partial (B_\varepsilon (x) \cap {\varOmega }) \cap \partial {\varOmega }$ is contained in $e \cup f_1 \cup f_2$.

Theorem A.5

Let ${\varOmega } \subset {\mathbb {R}}^2$ be a bounded Lipschitz polygon with vertices $A_j,\, j=1,\ldots ,J$, and edges ${\mathcal E}$. Let $M \subset \{A_1,\ldots , A_J\}$. Set ${\mathcal {A}}^\prime :=\{A_1,\ldots ,A_J\} \setminus M$ and fix $\varepsilon \in (0,1)$.

• (i) There is a sectorial neighborhood $S_{{\mathcal {A}}^\prime ,M,\widetilde{c}}$ of the vertices ${\mathcal {A}}^\prime$ and a constant $c \in (0,1)$ such that $S_{{\mathcal {A}}^\prime ,M,\widetilde{c}}$ is covered by balls $B_i:= \overline{B}_{c \, \mathrm{dist}(x_i,M)}(x_i)$ with centers $x_i \in {\mathcal A}^\prime$. Furthermore, the stretched balls $\widehat{B}_i:= \overline{B}_{(1+\varepsilon ) c\, \mathrm{dist}(x_i,M)}(x_i)$ are solid angles and satisfy a finite overlap property on ${\mathbb {R}}^2$.

• (ii) Fix a sectorial neighborhood ${\mathcal U} := S_{{\mathcal A}^\prime ,M,c^\prime }$ of the vertices ${\mathcal {A}}^\prime$. For each edge $e \in {\mathcal E}$, there is a sectorial neighborhood $S_{e,M,\widetilde{c}}$ and a constant $c \in (0,1)$ such that $S_{e,M,\widetilde{c}} \setminus {\mathcal {U}}$ is covered by balls $B_i = \overline{B}_{c \, \mathrm{dist}(x_i,M)}(x_i)$ whose centers $x_i$ are located on $e$. Furthermore, the stretched balls $\widehat{B}_i = \overline{B}_{(1+\varepsilon )c\, \mathrm{dist}(x_i,M)}(x_i)$ satisfy a finite overlap property on ${\mathbb {R}}^2$ and are such that each $\widehat{B}_i \cap {\varOmega }$ is a half-disk.

• (iii) Fix a sectorial neighbood ${\mathcal {U}}:= S_{{\mathcal {E}},M,c^\prime }$ of the edges ${\mathcal E}$. There is $c \in (0,1)$ such that ${\varOmega }\setminus {\mathcal U}$ is covered by balls $B_i = \overline{B}_{c \, \mathrm{dist}(x_i,M)}(x_i)$ such that the stretched balls $\widehat{B}_i = \overline{B}_{(1+\varepsilon ) c \, \mathrm{dist}(x_i,M)}(x_i)$ are completely contained in ${\varOmega }$ and satisfy a finite overlap property on ${\mathbb {R}}^2$.

Proof

The assertion (i) is almost trivial and only included to emphasize the structure of the arguments. Assertions (ii), (iii) follow from suitable applications of Lemmas A.3 and A.1. $\square$

The 3D variant of Theorem A.5 is formulated in Theorem A.6. We emphasize that the “singular” set $M$ need not be the union of all edges and vertices but can be just a subset. We also emphasize that it is not necessarily related to the notion of “singular set” in Definition 2.5, although it is used in this way. The key property of the covering balls is again such that the centers are either a) in ${\varOmega }$ (in which case the stretched ball is contained in ${\varOmega }$); or b) on a face (in which case the stretched ball $\widehat{B}_i$ is such that $\widehat{B}_i \cap {\varOmega }$ is a half-ball); or c) on an edge in which case $\widehat{B}_i \cap {\varOmega }$ is a dihedral angle (see Definition A.4); or d) in a vertex in which case $\widehat{B}_i \cap {\varOmega }$ is a solid angle (see Definition A.4).

Theorem A.6

Let ${\varOmega } \subset {\mathbb {R}}^3$ be a Lipschitz polyhedron with faces ${\mathcal {F}}$, edges ${\mathcal E}$, and vertices ${\mathcal A}$. Let $M_{\mathcal {A}} \subset {\mathcal A}$ and $M_{\mathcal E} \subset {\mathcal {E}}$. Let $M = \overline{M} = \overline{M_{\mathcal A} \cup M_{\mathcal E}}$ and fix $\varepsilon \in (0,1)$. Let ${\mathcal {A}}^\prime :=\{ A \in {\mathcal A}\,|\, A \not \in M\}$ be the vertices not in $M$ and ${\mathcal E}^\prime :=\{e \in {\mathcal {E}}\,|\, \overline{e} \cap M = \emptyset \}$ be the edges not abutting $M$. Then:

• (i) (non-singular vertices) There is a sectorial neighborhood $S_{{\mathcal A}^\prime ,M,\widetilde{c}}$ of the vertices in ${\mathcal A}^\prime$ and a constant $c \in (0,1)$ such that $S_{{\mathcal A}^\prime ,M,\widetilde{c}}$ is covered by balls $B_i:= \overline{B}_{c \,\mathrm{dist}(x_i,M)}(x_i)$ with centers $x_i \in {\mathcal A}^\prime$. Furthermore, the stretched balls $\widehat{B}_i:= \overline{B}_{(1+\varepsilon ) c \,\mathrm{dist}(x_i,M)}(x_i)$ are solid angles and satisfy a finite overlap property on ${\mathbb {R}}^3$.

• (ii) (non-singular edges) Fix a sectorial neighborhood ${\mathcal U} := S_{{\mathcal {A}}^\prime ,M,c^\prime }$ of ${\mathcal A}^\prime$. For each edge $e \in {\mathcal {E}}^\prime$, there is a sectorial neighborhood $S_{e,M,\widetilde{c}}$ and a constant $c \in (0,1)$ such that $S_{e,M,\widetilde{c}} \setminus {\mathcal U}$ is covered by balls $B_i = \overline{B}_{c \,\mathrm{dist}(x_i,M)}(x_i)$ whose centers $x_i$ are located on $e$. Furthermore, the stretched balls $\widehat{B}_i = \overline{B}_{(1+\varepsilon )c \,\mathrm{dist}(x_i,M)}(x_i)$ satisfy a finite overlap property on ${\mathbb {R}}^3$ and $\widehat{B}_i \cap {\varOmega }$ is a dihedral angle.

• (iii) (faces) Fix a sectorial neighbood ${\mathcal {U}}:= S_{{\mathcal {E}},M,c^\prime }$ of ${\mathcal {E}}$. There is a sectorial neighborhood $S_{{\mathcal {F}},M,\widetilde{c}}$ and a constant $c \in (0,1)$ such that $S_{{\mathcal {F}},M,\widetilde{c}}\setminus {\mathcal {U}}$ is covered by balls $B_i = \overline{B}_{c \,\mathrm{dist}(x_i,M)}(x_i)$ with centers $x_i \in \partial {\varOmega }$. Furthermore, the stretched balls $\widehat{B}_i = \overline{B}_{(1+\varepsilon ) c \,\mathrm{dist}(x_i,M)}(x_i)$ satisfy a finite overlap property on ${\mathbb {R}}^3$ and $\widehat{B}_i \cap {\varOmega }$ is a half-ball.

• (iv) (interior) Fix a sectorial neighbood ${\mathcal {U}}:= S_{{\mathcal {F}},M,c^\prime }$ of ${\mathcal {F}}$, where ${\mathcal {F}}$ is the set of faces. Then there is $c \in (0,1)$ such that ${\varOmega }\setminus {\mathcal {U}}$ is covered by balls $B_i = \overline{B}_{c \, \mathrm{dist}(x_i,M)}(x_i)$ with centers $x_i \in {\varOmega }$. Furthermore, the stretched balls $\widehat{B}_i = \overline{B}_{(1+\varepsilon ) c \,\mathrm{dist}(x_i,M)}(x_i)$ satisfy a finite overlap property on ${\mathbb {R}}^3$ and $\widehat{B}_i \subset {\varOmega }$.

Proof

Follows from Lemmas A.3 and A.1. $\square$