Abstract
The local behavior of the lowest order boundary element method on quasi-uniform meshes for Symm’s integral equation and the stabilized hyper-singular integral equation on polygonal/polyhedral Lipschitz domains is analyzed. We prove local a priori estimates in \(L^2\) for Symm’s integral equation and in \(H^1\) for the hyper-singular equation. The local rate of convergence is limited by the local regularity of the sought solution and the sum of the rates given by the global regularity and additional regularity provided by the shift theorem for a dual problem.
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1 Introduction
The boundary element method (BEM) for the discretization of boundary integral equations is an established numerical method for solving partial differential equations on (un)bounded domains. As an energy projection method, the Galerkin BEM is, like the finite element method (FEM), (quasi-)optimal in some global norm. However, often the quantity of interest is not the error on the whole domain, but rather a local error on part of the computational domain. For the FEM, the analysis of local errors goes back at least to [18]; advanced versions can be found in [10, 30]. For the Poisson problem, the local error estimates typically have the form
where u is the exact solution, \(u_h\) the finite element approximation from a space \(X_h\) of piecewise polynomials, and \(B_0 \Subset B_1\) are open subsets of \(\varOmega \) with \(R{:}{=}\text {dist}(B_0,\partial B_1)\). Thus, the local error in the energy norm is bounded by the local best approximation on a larger domain and the error in the weaker \(L^2\)-norm. The local best approximation allows for convergence rates limited only by the local regularity; the \(L^2\)-error is typically controlled with a duality argument and limited by the regularity of the dual problem as well as the global regularity of the solution. Therefore, if the solution is smoother locally, we can expect better rates of convergence for the local error.
Significantly fewer works study the local behavior of the BEM. The case of smooth two dimensional curves is treated in [5, 21, 28], in [27] three dimensional screen problems are studied, and [14] discusses local error estimates on polygons. [19, 20] provide estimates in the \(L^{\infty }\)-norm on smooth domains. Local error estimators for the BEM are presented in [23]. However, for the case of piecewise smooth geometries such as polygonal and polyhedral domains, sharp local error estimates that exploit the maximal (local) regularity of the solution are not available. Moreover, the analyses of [14, 21, 27, 28] are tailored to the energy norm and do not provide optimal local estimates in stronger norms, whereas [5] imposes additional global regularity.
In this article, we obtain sharp local error estimates for lowest order discretizations on quasi-uniform meshes for Symm’s integral equation in the \(L^2\)-norm and for the (stabilized) hyper-singular integral equation in the \(H^1\)-seminorm on polygonal/polyhedral domains. Structurally, the local estimates are similar to (1.1): The local error is bounded by a local best approximation error and a global error in a weaker norm. More precisely, our local convergence rates depend only on the local regularity and the sum of the rates given by the global regularity and the additional regularity of the dual problem on polygonal/polyhedral domains. Numerical examples show the sharpness of our analysis. As discussed in Remark 2.5 below, our results improve [21, 27, 28] as estimates in \(L^2\) (for Symm’s equation) and \(H^1\) (for the hyper-singular equation) are obtained there from local energy norm estimates with the aid of inverse estimates, thereby leading to a loss of \(h^{-1/2}\). In contrast, we avoid using an inverse inequality to go from the energy norm to a stronger norm.
The paper is structured as follows. We start with some notations and then present the main results for both Symm’s integral equation and the hyper-singular integral equation in Sect. 2. In Sects. 3 and 4 we are concerned with the proofs of these results. First, some technical preliminaries that exploit the additional regularity on piecewise smooth geometries to prove some improved a priori estimates for solutions of Poisson’s equation as well as for the boundary integral operators are presented. Then, we prove the main results, first for Symm’s equation, then for the stabilized hyper-singular equation. In principle, the proofs take ideas from [30], but important modifications of the arguments are necessary due to the nonlocal character of the integral operators. As in [30] a key ingredient are interior regularity estimates, which were provided recently in [11, 12], and to exploit some additional smoothing properties of commutators that arise in a localization step. Finally, Sect. 5 provides numerical examples that underline the sharpness of our theoretical local a priori estimates.
1.1 Notation on norms
For domains \(\omega \subset {\mathbb {R}}^d\), we define the integer order Sobolev spaces \(H^k(\omega )\), \(k \in {\mathbb {N}}_0\), in the standard way [15, p. 73ff]. The fractional Sobolev spaces \(H^{k + s}(\omega )\), \(k \in {\mathbb {N}}_0\), \(s \in (0,1)\) are defined by the Slobodeckii norm as described in [15, p. 73ff]. For open sets \(\omega = \cup _{i=1}^m \omega _i\) consisting of finitely many components of connectedness \(\omega _i\), the Sobolev spaces \(H^{k+s}(\omega )\) are understood in a piecewise way with norm \(\Vert u\Vert ^2_{H^{k+s}(\omega )}= \sum _i \Vert u\Vert ^2_{H^{k+s}(\omega _i)}\). The spaces \({\widetilde{H}}^s(\omega )\), \(s \ge 0\), consist of those function whose zero extension to \({\mathbb {R}}^d\) is in \(H^s({\mathbb {R}}^d)\). The spaces \(H^{-s}(\omega )\), \(s \ge 0\), are taken to be the dual space of \({\widetilde{H}}^s(\omega )\). We will make use of the fact that for bounded Lipschitz domains \(\omega \)
For Lipschitz domains \(\varOmega \subset {\mathbb {R}}^d\) with boundary \(\varGamma {:}{=} \partial \varOmega \), we define Sobolev spaces \(H^s(\varGamma )\) with \(s \in [0,1]\) as described in [15, p. 96ff] using local charts. For \(s > 1\), we define the spaces \(H^s(\varGamma )\) in a non-standard way: \(H^s(\varGamma )\) consists of those functions that have a lifting to \(H^{1/2+s}({\mathbb {R}}^d)\), and we define the norm \(\Vert \cdot \Vert _{H^s(\varGamma )}\) by
Correspondingly, for \(s>1\) there is a lifting operator
with the lifting property \( ({\mathcal {L}} u)|_{\varGamma } = u\), which is bounded by definition of the norm (1.3). The spaces \(H^{-s}(\varGamma )\), \(s \ge 0\), are the duals of \(H^s(\varGamma )\). Their norm is defined as
Remark 1.1
(equivalent norm definitions)
-
(i)
For \(s > 1\) an equivalent definition of the norm \(\Vert \cdot \Vert _{H^s(\varGamma )}\) in (1.3) would be to replace \(\Vert \cdot \Vert _{H^{s+1/2}({\mathbb R}^d)}\) with \(\Vert \cdot \Vert _{H^{s+1/2}(\varOmega )}\), i.e.,
$$\begin{aligned} \left\| u \right\| _{H^{s}(\varGamma )} := \inf \{ \left\| v \right\| _{H^{1/2+s}(\varOmega )}\,:\, v\in H^{1/2+s}(\varOmega ), v|_{\varGamma }=u \}. \end{aligned}$$This follows from the existence of the universal extension operator \(E:L^2(\varOmega ) \rightarrow L^2({\mathbb {R}}^d)\) described in [25, Chap. VI.3], which asserts that E is also a bounded linear operator \(H^{k}(\varOmega ) \rightarrow H^k({\mathbb {R}}^d)\) for any \(k \ge 0\).
-
(ii)
The trace operator \(\gamma _0: H^{s+1/2}({\mathbb {R}}^d) \rightarrow H^{s}(\varGamma )\) is a continuous operator for \(0< s < 1\) (cf. [15, Thm. 3.38], [22, Thm. 2.6.8], [17, Thm. 2.3]). [22, Thm. 2.6.11] (cf. also [15, Thm. 3.37], [17, Lem. 2.6]) assert the existence of a continuous lifting \({\mathcal {L}}\) in the range \(0< s < 1\) as well so that (1.3) is an equivalent norm for \(0< s < 1\) as well.
-
(iii)
For polygonal (in 2D) and polyhedral (in 3D) Lipschitz domains the spaces \(H^{s}(\varGamma )\) in the range \(s \in (1,3/2)\) can be characterized alternatively as follows: Let \(\varGamma _i\), \(i=1,\ldots ,N\), be the affine pieces of \(\varGamma \), which may be identified with an interval (for the 2D case) or a polygon (for the 3D case). Then
$$\begin{aligned} u \in H^s(\varGamma ) \quad \Longleftrightarrow \quad u|_{\varGamma _i} \in H^s(\varGamma _i) \quad \forall i \in \{1,\ldots ,N\} \quad \text{ and } u \in C^0(\varGamma ). \end{aligned}$$(1.5)The equivalence (1.5) gives rise to yet another norm equivalence for the space \(H^s(\varGamma )\), namely, \(\Vert u\Vert _{H^s(\varGamma )} \sim \sum _{i=1}^N \Vert u\Vert _{H^s(\varGamma _i)}\). The condition \(u \in C^0(\varGamma )\) is a compatibility condition. More generally, for \(s > 3/2\) similar, more complicated compatibility conditions can be formulated to describe the space \(H^s(\varGamma )\) in terms of piecewise Sobolev spaces. \(\square \)
We will also need local norms on the boundary. For an open subset \(\varGamma _0 \subset \varGamma \) and \(s \ge 0\), we define local negative norms by
In the following, we write \(\gamma _0^\mathrm{int}\) for the interior trace operator, i.e., the trace operator from the inside of the domain and \(\gamma _0^\mathrm{ext}\) for the exterior trace operator. For the jump of the trace of a function u we use the notation \([\gamma _0 u] {:}{=} \gamma _0^\mathrm{ext} u-\gamma _0^\mathrm{int} u\). In order to shorten notation, we write \(\gamma _0\) for the trace, if the interior and exterior trace are equal, i.e., \([\gamma _0 u]=0\). We denote the interior and exterior conormal derivative by \(\gamma _1^\mathrm{int} u {:}{=} \gamma _0^\mathrm{int}\nabla u \cdot n\), \(\gamma _1^\mathrm{ext} u {:}{=} \gamma _0^\mathrm{ext}\nabla u \cdot n\), where n denotes the normal vector pointing into \({\mathbb R}^d {\setminus } \varOmega \). The jump of the normal derivative across the boundary is defined by \([\partial _n u]{:}{=} \gamma _1^\mathrm{ext} u- \gamma _1^\mathrm{int} u\), and we write \(\partial _n u\) for the normal derivative if \([\partial _n u] = 0\).
We will call axis-parallel squares/cubes “boxes”.
2 Main results
We study bounded Lipschitz domains \(\varOmega \subset {\mathbb {R}}^d\), \(d \ge 2\) with polygonal/polyhedral boundary \(\varGamma :=\partial \varOmega \).
2.1 Symm’s integral equation
The elliptic shift theorem for the Dirichlet problem is valid in a range that is larger than for general Lipschitz domains. We characterize this extended range by a parameter \(\alpha _D \in (0,1/2)\) that will pervade most of the estimates of the present work. It is defined by the following assumption:
Assumption 2.1
\(\varOmega \subset {\mathbb {R}}^d\), \(d\ge 2\), is a bounded Lipschitz domain whose boundary consists of finitely many affine pieces (i.e., \(\varOmega \) is the intersection of finitely many half-spaces). \(R_\varOmega > 0\) is such that the open ball \(B_{R_{\varOmega }}(0) \subset {\mathbb {R}}^d\) of radius \(R_\varOmega \) that is centered at the origin contains \({\overline{\varOmega }}\). The parameter \(\alpha _D \in (0,1/2)\) is such that for every \(\varepsilon \in (0,\alpha _D]\) there is \(C_\varepsilon > 0\) such that the a priori bound
holds, where \(u{:}{=} Tf \in H^1(B_{R_\varOmega }(0){\setminus }\varGamma )\) denotes the solution of
Recall that the norms \(\left\| \cdot \right\| _{H^{s}(B_{R_{\varOmega }}(0)\backslash \varGamma )}\), \(s>0\) are understood as the sum of the norm on \(\varOmega \) and \(B_{R_{\varOmega }}(0)\backslash \overline{\varOmega }\), i.e.,
Remark 2.2
The condition on the parameter \(\alpha _D\) in Assumption 2.1 can be described in terms of two Dirichlet problems, one posed on \(\varOmega \) and one posed on \(B_{R_\varOmega }(0) {\setminus } {\overline{\varOmega }}\). For each of these two domains, a shift theorem is valid, and \(\alpha _D\) is determined by the more stringent of the two conditions. It is worth stressing that the type of boundary condition on \(\partial B_{R_{\varOmega }}(0)\) is not essential in view of the smoothness of \(\partial B_{R_{\varOmega }}(0)\) and 
.
In the case \(d = 2\) the parameter \(\alpha _D\) is determined by the extremal angles of the polygon \(\varOmega \). Specifically, let \(0< \omega _j < 2 \pi \), \(j=1,\ldots ,J\), be the interior angles of the polygon \(\varOmega \). Then, Assumption 2.1 is valid for any \(\alpha _D >0 \) that satisfies
(Note that \(\omega _j \ne \pi \) for all j so that the right inequality is indeed strict.) \(\square \)
We consider Symm’s integral equation in its weak form: Given \( f \in H^{1/2}(\varGamma )\) find \(\phi \in H^{-1/2}(\varGamma )\) such that
Here, the single-layer operator V is given by
where, with the surface measure \(|S^{d-1}|\) of the Euclidean sphere in \({\mathbb {R}}^d\), we set
The single layer operator V is a bounded linear operator in \(L(H^{-1/2+s}(\varGamma ),H^{1/2+s}(\varGamma ))\) for \(\left| s \right| \le \frac{1}{2}\), [22, Thm. 3.1.16]. It is elliptic for \(s=0\) with the usual proviso for \(d=2\) that 
, which we may assume by scaling.
Let \(\mathcal {T}_h = \{T_1,\dots ,T_N\}\) be a quasi-uniform, regular and \(\gamma \)-shape regular triangulation of the boundary \(\varGamma \) with mesh-width 
. By \(S^{0,0}(\mathcal {T}_h){:}{=}\{u \in L^2(\varGamma ):u|_{T_j}\; \text {is constant} \,\forall T_j \in \mathcal {T}_h\}\) we denote the space of piecewise constant functions on the mesh \(\mathcal {T}_h\). The Galerkin formulation of (2.3) reads: Find \(\phi _h \in S^{0,0}(\mathcal {T}_h)\) such that
The following theorem is one of the main results of this paper. It estimates the Galerkin error in the \(L^2\)-norm on a subdomain by the local best approximation error in \(L^2\) on a slightly larger subdomain and the global error in a weaker norm.
Theorem 2.3
Let Assumption 2.1 hold and let \(\mathcal {T}_h\) be a quasi-uniform, \(\gamma \)-shape regular triangulation. Let \(\phi \in H^{-1/2}(\varGamma )\) and \(\phi _h \in S^{0,0}(\mathcal {T}_h)\) satisfy the Galerkin orthogonality condition
Let \(\varGamma _0\), \({\widehat{\varGamma }}\) be open subsets of \(\varGamma \) with \(\varGamma _0\subset {\widehat{\varGamma }} \subsetneq \varGamma \) and 
. Let h be such that \(C_{\alpha _D}\frac{h}{R}\le \frac{1}{12}\) with a fixed constant \(C_{\alpha _D}\) depending only on \(\alpha _D\). Assume that \(\phi \in L^{2}({\widehat{\varGamma }})\). Then, we have
The constant \(C>0\) depends only on \(\varGamma ,\varGamma _0,{\widehat{\varGamma }},d,R,\) and the \(\gamma \)-shape regularity of \(\mathcal {T}_h\).
If we additionally assume higher local regularity as well as some (low) global regularity of the solution \(\phi \), this local estimate implies that the local error converges faster than the global error, which is stated in the following corollary.
Corollary 2.4
Let the assumptions of Theorem 2.3 be fulfilled. Let \(\widetilde{\varGamma }\subset \varGamma \) be a subset with \({\widehat{\varGamma }}\subsetneq \widetilde{\varGamma }\) and 
. Additionally, assume \(\phi \in H^{-1/2+\alpha }(\varGamma ) \cap H^{\beta }(\widetilde{\varGamma })\) with \(\alpha \ge 0\), \(\beta \in [0,1]\). Then, we have
with a constant \(C>0\) depending only on \(\varGamma ,\varGamma _0,{\widehat{\varGamma }},\widetilde{\varGamma },d,R,\alpha ,\beta \), and the \(\gamma \)-shape regularity of \(\mathcal {T}_h\).
In the results of [18, 30] singularities far from the domain of interest still have a weak influence on the local convergence of the FEM. Corollary 2.4 shows that this is similar in the BEM: The a priori estimate shows the effect of singularities of the solution (represented by \(\alpha \)) and those induced by the geometry (represented by \(\alpha _D\)) affect the local convergence.
Remark 2.5
In comparison to [27], Corollary 2.4 gives a better result for the rate of convergence of the local error in the case where the convergence is limited by the global error in the weaker norm. More precisely, for the case \(\phi \in H^{1/2}(\widetilde{\varGamma }) \cap L^2(\varGamma )\), [27] obtains the local rate of 1 / 2, which coincides with our local rate. However, if \(\phi \in H^{1}(\widetilde{\varGamma })\), we obtain rate 1 in the \(L^2\)-norm, whereas the rate in [27] remains 1 / 2. \(\square \)
Remark 2.6
Even for smooth functions f, the solution \(\phi \) of (2.3) is, in general, not better than \(H^{\alpha }(\varGamma )\) with \(\alpha = \frac{1}{2} + \alpha _D\). Recall from Remark 2.2 that \(\alpha _D\) is determined by the mapping properties for both the interior and the exterior Dirichlet problem. A special situation therefore arises if Symm’s integral equation is obtained from reformulating an interior (or exterior) Dirichlet problem. To be specific, consider again the case \(d = 2\) of a polygon \(\varOmega \) with interior angles \(\omega _j\), \(j=1,\ldots ,J\). We rewrite the boundary value problem \(-\varDelta u = 0\) in \(\varOmega \) with \(u|_\varGamma = g\) as the integral equation
for the unknown function \(\phi = \partial _n u\) with the double layer operator K defined by
Then, \(\phi \in H^\alpha (\varGamma )\) for any \(\alpha \) with \(\alpha < 1/2 + \min _j \frac{\pi }{\omega _j}\). \(\square \)
2.2 The hyper-singular integral equation
For the Neumann problem, we assume an extended shift theorem as well.
Assumption 2.7
\(\varOmega \subset {\mathbb {R}}^d\), \(d\ge 2\), is a bounded Lipschitz domain whose boundary consists of finitely many affine pieces (i.e., \(\varOmega \) is the intersection of finitely many half-spaces). \(R_\varOmega > 0\) is such that the open ball \(B_{R_{\varOmega }}(0) \subset {\mathbb {R}}^d\) contains \({\overline{\varOmega }}\). The parameter \(\alpha _N \in (0,\alpha _D]\), where \(\alpha _D\) is the parameter from Assumption 2.1, is such that for every \(\varepsilon \in (0,\alpha _N]\) there is \(C_\varepsilon > 0\) such that for all \(f \in H^{-1/2+\varepsilon }(B_{R_\varOmega }(0){\setminus }\varGamma )\) and \(g \in H^{\varepsilon }(\varGamma )\) with \(\int _{\varOmega }f +\int _{\varGamma } g = 0\) the a priori bound
holds, where \(u{:}{=} Tf \in H^1(B_{R_\varOmega }(0){\setminus }\varGamma )\) denotes the solution of
The condition on the parameter \(\alpha _N\) again can be described in terms of two problems, a pure Neumann problem posed in \(\varOmega \), for which we need a compatibility condition, and a mixed Dirichlet–Neumann problem posed on \(B_{R_\varOmega }(0) \backslash \overline{\varOmega }\), which is uniquely solvable without the need to impose a solvability condition for f, g.
The parameter \(\alpha _N\) again depends only on the geometry and the corners/edges that induce singularities. In fact, on polygonal domains, i.e., \(d=2\), \(\alpha _D = \alpha _N\), see, e.g., [9].
Studying the inhomogeneous Neumann boundary value problem \(-\varDelta u = 0\), \(\partial _n u = g\), leads to the boundary integral equation of finding \(\varphi \in H^{1/2}(\varGamma )\) such that \(W\varphi = f\) with \(f \in H^{-1/2}(\varGamma )\) satisfying the compatibility condition \(\left\langle f,1 \right\rangle _{L^2(\varGamma )} = 0\), and the hyper-singular integral operator \(W \in L(H^{1/2}(\varGamma ),H^{-1/2}(\varGamma ))\) defined by
We additionally assume that \(\varGamma \) is connected, so that the hyper-singular integral operator has a kernel of dimension one consisting of the constant functions. Therefore, the boundary integral equation is not uniquely solvable. Employing the constraint \(\left\langle \varphi ,1 \right\rangle _{L^2(\varGamma )} = 0\) leads to the stabilized variational formulation
which has a unique solution \(\varphi \in H^{1/2}(\varGamma )\), see, e.g., [26]. For the Galerkin discretization we employ lowest order test and trial functions in \(S^{1,1}(\mathcal {T}_h){:}{=}\{u \in H^1(\varGamma ):u|_{T_j} \in \mathcal {P}_1 \,\forall T_j \in \mathcal {T}_h\}\), which leads to the discrete variational problem of finding \(\psi _h \in S^{1,1}(\mathcal {T}_h)\) such that
The following theorem is the analog of Theorem 2.3 for the hyper-singular integral equation. The local error in the \(H^1\)-seminorm is estimated by the local best approximation error and the global error in a weak norm.
Theorem 2.8
Let Assumption 2.7 hold and let \(\mathcal {T}_h\) be a quasi-uniform, \(\gamma \)-shape regular triangulation. Let \(\varphi \in H^{1/2}(\varGamma )\) and \(\varphi _h \in S^{1,1}(\mathcal {T}_h)\) satisfy the Galerkin orthogonality condition
Let \(\varGamma _0\), \({\widehat{\varGamma }}\) be open subsets of \(\varGamma \) with \(\varGamma _0\subset {\widehat{\varGamma }} \subsetneq \varGamma \) and 
. Let h be such that \(C_{\alpha _N}\frac{h}{R}\le \frac{1}{12}\) with a fixed constant \(C_{\alpha _N}\) depending only on \(\alpha _N\). Assume that \(\varphi \in H^{1}({\widehat{\varGamma }})\). Then, we have
The constant \(C>0\) depends only on \(\varGamma ,\varGamma _0,{\widehat{\varGamma }},d,R,\) and the \(\gamma \)-shape regularity of \(\mathcal {T}_h\).
Again, assuming additional regularity, the local estimate of Theorem 2.8 leads to a higher rate of local convergence of the BEM for the stabilized hyper-singular integral equation.
Corollary 2.9
Let the assumptions of Theorem 2.8 be fulfilled. Let \(\widetilde{\varGamma }\subset \varGamma \) be a subset with \({\widehat{\varGamma }}\subsetneq \widetilde{\varGamma }\), 
. Additionally, assume \(\varphi \in H^{1/2+\alpha }(\varGamma ) \cap H^{1+\beta }(\widetilde{\varGamma })\) with \(\alpha \ge 0\), \(\beta \in [0,1]\). Then, we have
with a constant \(C>0\) depending only on \(\varGamma ,\varGamma _0,{\widehat{\varGamma }},\widetilde{\varGamma },d,R,\alpha ,\beta \), and the \(\gamma \)-shape regularity of \(\mathcal {T}_h\).
3 Shift theorems
The following two sections are dedicated to the proofs of Theorem 2.3 and Corollary 2.4 for Symm’s integral equation as well as Theorem 2.8 and Corollary 2.9 for the hyper-singular integral equation. We start with some technical results that are direct consequences of the assumed shift theorems from Assumption 2.1 for the Dirichlet problem and Assumption 2.7 for the Neumann problem. The shift theorem of Assumption 2.1 implies the following shift theorem for Dirichlet problems:
Lemma 3.1
Let the shift theorem from Assumption 2.1 hold and let u be the solution of the inhomogeneous Dirichlet problem \(-\varDelta u = 0\) in \(B_{R_{\varOmega }}(0)\backslash \varGamma \), \(\gamma _0 u = g\) on \(\varGamma \cup \partial B_{R_{\varOmega }}(0)\) for some \(g\in H^{1/2}(\varGamma \cup \partial B_{R_{\varOmega }}(0))\).
-
(i)
There is a constant \(C>0\) depending only on \(\varOmega \) and \(\alpha _D\) such that
$$\begin{aligned} \left\| u \right\| _{H^{1/2-\alpha _D}(B_{R_{\varOmega }}(0)\backslash \varGamma )}\le C \left\| g \right\| _{H^{-\alpha _D}(\varGamma \cup \partial B_{R_{\varOmega }}(0))}. \end{aligned}$$(3.1) -
(ii)
Let \(\varepsilon \in (0,\alpha _D]\) and \(B\subset B' \subset B_{R_{\varOmega }}(0)\) be nested subdomains with
. Let \(\eta \in C^{\infty }({\mathbb {R}}^d)\) be a cut-off function satisfying \(\eta \equiv 1\) on \(B\cap \varGamma \), 
, and 
for \(k \in \{0,1,2\}\). Assume \(\eta g\in H^{1+\varepsilon }(\varGamma )\). Then $$\begin{aligned} \left\| u \right\| _{H^{3/2+\varepsilon }(B\backslash \varGamma )} \le C\left( \left\| u \right\| _{H^1(B'\backslash \varGamma )} +\left\| \eta g \right\| _{H^{1+\varepsilon }(\varGamma )}\right) . \end{aligned}$$(3.2)Here, the constant \(C>0\) additionally depends on
.
. Let 
, and 
for 
.Proof:
Proof of (i): Let v solve \(-\varDelta v = w\) in \(B_{R_{\varOmega }}(0)\backslash \varGamma \), \(\gamma _0 v = 0\) on \(\varGamma \cup \partial B_{R_{\varOmega }}(0)\) for \(w \in H^{-1/2+\alpha _D}(B_{R_{\varOmega }}(0)\backslash \varGamma )\). Then, in view of (1.2), we have
Integration by parts on \(\varOmega \) and \(B_{R_{\varOmega }}(0)\backslash \overline{\varOmega }\) and the boundary condition \(\gamma _0 v = 0\) lead to
We split the polygonal/polyhedral boundary \(\varGamma = \bigcup _{\ell =1}^m \overline{\varGamma _{\ell }}\) into its (smooth) faces \(\varGamma _{\ell }\) and prolong each face \(\varGamma _{\ell }\) to the hyperplane \(\varGamma _{\ell }^{\infty }\), which decomposes \({\mathbb {R}}^d\) into two half spaces \(\varOmega _{\ell }^{\pm }\). Let \(\chi _{\ell } \in L^2(\varGamma )\) be the characteristic function for \(\varGamma _{\ell }\). Since the normal vector on a face does not change, we may use the trace estimate (note: \(0<\alpha _D < 1/2\)) facewise, to estimate
As the boundary \(\partial B_{R_{\varOmega }}(0)\) is smooth, standard elliptic regularity yields \(\left\| \partial _n v \right\| _{H^{\alpha _D}(\partial B_{R_{\varOmega }}(0))} \lesssim \left\| v \right\| _{H^{3/2+\alpha _D}(B_{R_{\varOmega }}(0)\backslash \varGamma )}\). This leads to
Proof of (ii): With the lifting operator \({\mathcal {L}}:H^{1+\varepsilon }(\varGamma )\rightarrow H^{3/2+\varepsilon }(B_{R_{\varOmega }}(0)\backslash \varGamma )\) from (1.4), the function \({\widetilde{u}}{:}{=}\eta ^2 u - \eta {\mathcal {L}}(\eta g)\) satisfies
With the shift theorem from Assumption 2.1 we get
which proves the second statement. \(\square \)
The following lemma collects mapping properties of the single-layer operator V that exploits the present setting of piecewise smooth geometries:
Lemma 3.2
Define the single layer potential \({\widetilde{V}}\) by
-
(i)
The single layer potential \({\widetilde{V}}\) is a bounded linear operator from \(H^{-1/2+s}(\varGamma )\) to \(H^{1+s}(B_{R_\varOmega }(0)\backslash \varGamma )\) for \(-1/2\le s < 1\).
-
(ii)
The single-layer operator V is a bounded linear operator from \(H^{-1/2+s}(\varGamma )\) to \(H^{1/2+s}(\varGamma )\) for \(-1/2 \le s < 1\).
-
(iii)
The adjoint double-layer operator \(K'\) is a bounded linear operator from \(H^{-1/2+s}(\varGamma )\) to \(H^{-1/2+s}(\varGamma )\) for \(-1/2 \le s < 1\).
Proof:
Proof of (i): The case \(s \in (-1/2,1/2)\) is shown in [22, Thm. 3.1.16], and for \(s=-\frac{1}{2}\) we refer to [29]. For the case \(s \in [1/2,1)\), we exploit that \(\varGamma \) is piecewise smooth. We split the polygonal/polyhedral boundary \(\varGamma = \bigcup _{\ell =1}^m \overline{\varGamma _{\ell }}\) into its (smooth) faces \(\varGamma _{\ell }\). Let \(\chi _{\ell } \in L^2(\varGamma )\) be the characteristic function for \(\varGamma _{\ell }\). Then, for \(\varphi \in H^{-1/2+s}(\varGamma )\), we have \({\widetilde{V}} \varphi =\sum _{\ell =1}^m{\widetilde{V}}(\chi _{\ell }\varphi )\). We prolong each face \(\varGamma _{\ell }\) to the hyperplane \(\varGamma _{\ell }^{\infty }\), which decomposes \({\mathbb {R}}^d\) into two half spaces \(\varOmega _{\ell }^{\pm }\). Due to \(s <1\), we have \(\chi _{\ell }\varphi \in H^{-1/2+s}(\varGamma _{\ell }^{\infty })\). Since the half spaces \(\varOmega _{\ell }^{\pm }\) have smooth boundaries, we may use the mapping properties of \({\widetilde{V}}\) on smooth geometries, see, e.g., [15, Thm. 6.13] to estimate
Proof of (ii): The case \(-1/2 \le s \le 1/2\) is taken from [22, Thm. 3.1.16]. For \(s \in (1/2,1)\) the result follows from part (i) and the definition of the norm \(\Vert \cdot \Vert _{H^s(\varGamma )}\) given in (1.3).
Proof of (iii): The case \(-1/2 \le s \le 1/2\) is taken from [22, Thm. 3.1.16]. With \(K' = \partial _n {\widetilde{V}}-\frac{1}{2}\mathrm {Id}\) the case \(s \in (1/2,1)\) follows from part (i) and a facewise trace estimate (3.3) since
\(\square \)
In addition to the single layer operator V, we will need to understand localized versions of these operators, i.e., the properties of commutators. For a smooth cut-off function \(\eta \), we define the commutators
Since the singularity of the Green’s function at \(x=y\) is smoothed by \(\eta (x)-\eta (y)\), we expect that the commutators \(C_{\eta }\), \(C_{\eta }^{\eta }\) have better mapping properties than the single-layer operator; this is stated in the following lemma.
Lemma 3.3
Let \(\eta \in C^\infty _0({\mathbb {R}}^d)\) be fixed.
-
(i)
Let \(s\in (-1/2,1/2)\). The commutator \(C_{\eta }\) can be extended in a unique way to a bounded linear operator \(C_{\eta }:H^{-1+s}(\varGamma )\rightarrow H^{1+s}(\varGamma )\). The continuity constant depends only on \(\Vert \eta \Vert _{W^{1,\infty }({\mathbb {R}}^d)}\), \(\varOmega \), and s. Furthermore, the operator is skew-symmetric (with respect to the \(L^2(\varGamma )\)-inner product).
-
(ii)
The commutator \(C_{\eta }^{\eta }\) is a symmetric and continuous mapping \(C_{\eta }^{\eta }:H^{-1-\alpha _D}(\varGamma ) \rightarrow H^{1+\alpha _D}(\varGamma )\). Here, the continuity constant depends only on \(\Vert \eta \Vert _{W^{1,\infty }({\mathbb {R}}^d)}\), \(\varOmega \), and the constants appearing in Assumption 2.1.
Proof:
Proof of (i): 1. step: We show the boundedness for the case \(0< s < 1/2\). Let \(\widehat{\phi } \in H^{-1+s}(\varGamma )\), and set
Since the volume potential \({\widetilde{V}} \widehat{\phi } \) is harmonic and in view of the jump relations \([\gamma _0 {\widetilde{V}}\phi ] = 0\), \([\partial _n {\widetilde{V}}\phi ] = -\phi \) satisfied by \({\widetilde{V}}\), cf. [22, Thm. 3.3.1], we have
We may write \(u = {\mathcal {N}}(2\nabla \eta \cdot \nabla {\widetilde{V}}\widehat{\phi } + \varDelta \eta {\widetilde{V}}\widehat{\phi })\) with the Newton potential
since u and \({\mathcal {N}}(2\nabla \eta \cdot \nabla {\widetilde{V}}\widehat{\phi } + \varDelta \eta {\widetilde{V}}\widehat{\phi })\) have the same decay for \(\left| x \right| \rightarrow \infty \). The mapping properties of the Newton potential (see, e.g., [22, Thm. 3.1.2]), as well as the mapping properties of \({\widetilde{V}}\) of Lemma 3.2, (i) provide
The definition of \(C_{\eta }\) and the definition of the norm \(\left\| \cdot \right\| _{H^{1+s}(\varGamma )}\) from (1.3) prove the mapping properties of \(C_{\eta }\) for \(0< s < 1/2\).
The mapping properties of the Newton potential ( see, e.g., [22, Thm. 3.1.2]) also lead to
2. step: Since V is symmetric, we have for arbitrary \(\widehat{\phi }\), \(\psi \in H^{-1/2}(\varGamma )\)
With the mapping property \(C_{\eta }: H^{-1+s}(\varGamma ) \rightarrow H^{1+s}(\varGamma )\) for \(0<s<1/2\), we see that the right-hand side of this equation extends to a bounded linear functional on \(H^{-1-s}(\varGamma )\), which proves the mapping properties for the case \(-1/2<s<0\).
A similar computation proves the symmetry of the commutator \(C_{\eta }^{\eta }\) asserted in (ii).
3. step: We have \(C_{\eta }: H^{-1+s}(\varGamma )\rightarrow H^{1+s}(\varGamma )\) for \(s \in (-1/2,1/2)\backslash \{0\}\). An interpolation argument extends the boundedness to the remaining case \(s=0\).
Proof of (ii): Let
Since
the function v solves
Again, the function v and the Newton potential of the right-hand side of (3.12) have the same decay for \(\left| x \right| \rightarrow \infty \), and the mapping properties of the Newton potential as well as the previous estimate (3.10) for \({\widetilde{C}}_{\eta }\widehat{\phi }\) provide
We apply Lemma 3.1 to \({\widetilde{V}}\widehat{\phi }\). Since 
, we have that \({\widetilde{V}}\widehat{\phi }\) is smooth on \(\partial B_{R_{\varOmega }}(0)\), and we can estimate this term by an arbitrary negative norm of \(\widehat{\phi }\) on \(\varGamma \) to obtain
The mapping properties of V of Lemma 3.2, (ii) and the symmetry of V imply
Inserting this in (3.14) leads to \( \left\| v \right\| _{H^{3/2+\alpha _D}(B_{R_{\varOmega }}(0)\backslash \varGamma )} \lesssim \left\| \widehat{\phi } \right\| _{H^{-1-\alpha _D}(\varGamma )}, \) which, together with the definition of the \(H^{1+\alpha _D}(\varGamma )\)-norm in (1.3), proves the lemma. \(\square \)
The shift theorem for the Neumann problem from Assumption 2.7 implies the following shift theorem.
Lemma 3.4
Let Assumption 2.7 be valid, and let u be the solution of the inhomogeneous problems
where \(g_N\in H^{-1/2}(\varGamma )\) with \(\left\langle g_N,1 \right\rangle _{L^2(\varGamma )} = 0\), and \(g_D\in H^{1/2}(\partial B_{R_{\varOmega }}(0))\).
-
(i)
There is a constant \(C>0\) depending only on \(\varOmega \) and \(\alpha _N\) such that
$$\begin{aligned} \left\| u \right\| _{H^{1/2-\alpha _N}(B_{R_{\varOmega }}(0)\backslash \varGamma )}\le C \left( \left\| g_N \right\| _{H^{-1-\alpha _N}(\varGamma )} + \left\| g_D \right\| _{H^{-\alpha _N}(\partial B_{R_{\varOmega }}(0))}\right) . \end{aligned}$$(3.15) -
(ii)
Let \(\varepsilon \in (0,\alpha _N]\). Let \(B\subset B' \subset B_{R_{\varOmega }}(0)\) be nested subdomains with
and let \(\eta \in C_0^{\infty }({\mathbb {R}}^d)\) satisfy \(\eta \equiv 1\) on \(B\cap \varGamma \), 
, and 
for \(k \in \{0,1,2\}\). Assume \(\eta g_N\in H^{\varepsilon }(\varGamma )\). Then $$\begin{aligned} \left\| u \right\| _{H^{3/2+\varepsilon }(B\backslash \varGamma )} \le C\left( \left\| u \right\| _{H^1(B'\backslash \varGamma )} +\left\| \eta g_N \right\| _{H^{\varepsilon }(\varGamma )}\right) . \end{aligned}$$(3.16)Here, the constant \(C>0\) depends on \(\varOmega , \alpha _N\), and
.
and let 
, and 
for 
.Proof:
Proof of (i): For \(w \in H^{-1/2+\alpha _N}(B_{R_{\varOmega }}(0)\backslash \varGamma )\) and \(\overline{w}:=\frac{1}{\left| \varOmega \right| }\left\langle w,1 \right\rangle _{L^2(\varOmega )}\), let v solve
Then, with \(\left\langle u,1 \right\rangle _{L^2(\varOmega )} = 0\), we have
Integration by parts on \(\varOmega \) and \(B_{R_{\varOmega }}(0)\backslash \overline{\varOmega }\) and the boundary conditions of v lead to
The definition of the norm (1.3) implies
and the same estimate holds for \(\gamma _0^\mathrm{ext} v\). Since \(\partial B_{R_{\varOmega }}(0)\) is smooth, we may estimate with the trace inequality
This leads to
Proof of (ii): Since \(\eta \equiv 0\) on \(\partial B_{R_{\varOmega }}(0)\), the function \({\widetilde{u}}{:}{=}\eta u\) satisfies
The shift theorem from Assumption 2.7 and the trace inequality \(\left\| (\partial _n \eta ) \gamma _0^\mathrm{int}u \right\| _{H^{1/2}(\varGamma )} \lesssim \left\| u \right\| _{H^1(B'\backslash \varGamma )}\) provide
which proves the second statement. \(\square \)
The following lemma collects mapping properties of the double-layer operator K and the hyper-singular operator W that exploit the present setting of piecewise smooth geometries:
Lemma 3.5
Define the double layer potential \({\widetilde{K}}\) by
-
(i)
The double layer potential \({\widetilde{K}}\) is a bounded linear operator from \(H^{1/2+s}(\varGamma )\) to \(H^{1+s}(B_{R_\varOmega }(0)\backslash \varGamma )\) for \(-1/2\le s \le 1/2+\alpha _N\).
-
(ii)
The double layer operator K is a bounded linear operator from \(H^{1/2+s}(\varGamma )\) to \(H^{1/2+s}(\varGamma )\) for \(-1/2 \le s \le 1/2+\alpha _N\).
-
(iii)
The hyper singular operator W is a bounded linear operator from \(H^{1/2+s}(\varGamma )\) to \(H^{-1/2+s}(\varGamma )\) for \(-1/2-\alpha _N \le s \le 1/2+\alpha _N\).
Proof:
Proof of (i): With the mapping properties of the single layer potential \({\widetilde{V}}\in L(H^{-1/2+s}(\varGamma ),H^{1+s}(B_{R_\varOmega }(0)\backslash \varGamma ))\) from Lemma 3.2, the mapping properties of the solution operator of the Dirichlet problem from Assumption 2.1 (\(T:H^{1/2+s}(\varGamma )\rightarrow H^{1+s}(B_{R_\varOmega }(0)\backslash \varGamma )\)), and the assumption \(\alpha _N \le \alpha _D\), the mapping properties of \({\widetilde{K}}\) follow from Green’s formula by expressing \({\widetilde{K}}\) in terms of \({\widetilde{V}}\), T, and the Newton potential \({\mathcal {N}}\). For details, we refer to [22, Thm. 3.1.16], where the case \(s \in (-1/2,1/2)\) is shown.
Proof of (ii): The case \(-1/2 \le s \le 1/2\) is taken from [22, Thm. 3.1.16]. For \(s \in (1/2,1/2+\alpha _N]\) the result follows from part (i), the definition of the norm \(\Vert \cdot \Vert _{H^{s+1/2}(\varGamma )}\) given in (1.3), and \(K = \gamma _0^\mathrm{int}{\widetilde{K}}+\frac{1}{2}\mathrm {Id}\).
Proof of (iii): The case \(-1/2 \le s \le 1/2\) is taken from [22, Thm. 3.1.16]. Since \(W = -\partial _n {\widetilde{K}}\), we get with a facewise trace estimate as in the proof of Lemma 3.1, estimate (3.3), that
which finishes the proof for the case \(s \in (1/2,1/2+\alpha _N]\). With the symmetry of W, the case \(s \in [-1/2-\alpha _N,-1/2)\) follows. \(\square \)
For a smooth function \(\eta \), we define the commutators
By the mapping properties of W, both operators map \(H^{1/2}(\varGamma ) \rightarrow H^{-1/2}(\varGamma )\). However, \({\mathcal {C}}_{\eta }\) is in fact an operator of order 0 and \({\mathcal {C}}_{\eta }^{\eta }\) is an operator of positive order:
Lemma 3.6
Fix \(\eta \in C_0^\infty ({\mathbb {R}}^d)\).
-
(i)
Let \(s \in (-1/2,1/2)\). Then, the commutator \({\mathcal {C}}_\eta \) can be extended in a unique way to a bounded linear operator \(H^s(\varGamma ) \rightarrow H^s(\varGamma )\), which satisfies
$$\begin{aligned} \Vert {\mathcal C}_\eta \varphi \Vert _{H^{s}(\varGamma )} \le C \Vert \varphi \Vert _{H^s(\varGamma )} \qquad \forall \varphi \in H^s(\varGamma ). \end{aligned}$$(3.20)The constant C depends only on \(\Vert \eta \Vert _{W^{1,\infty }({\mathbb {R}}^d)}\), \(\varOmega \), and s. Furthermore, the operator is skew-symmetric (with respect to the extended \(L^2\)-inner product).
-
(ii)
The commutator \({\mathcal {C}}_{\eta }^{\eta }\) is a symmetric and continuous mapping \({\mathcal {C}}_{\eta }^{\eta }:H^{-\alpha _N}(\varGamma ) \rightarrow H^{\alpha _N}(\varGamma )\). The continuity constant depends only on \(\Vert \eta \Vert _{W^{1,\infty }({\mathbb {R}}^d)}\), \(\varOmega \), and the constants appearing in Assumption 2.7.
Proof:
Proof of (i): 1. step: We show (3.20) for the range \(0< s < 1/2\). For \(\varphi \in H^{1/2}(\varGamma )\), consider the operator
with the single layer potential \({\widetilde{V}}\) and the double layer potential \({\widetilde{K}}\) from (3.17). Using the jump conditions \([\gamma _0 {\widetilde{V}}\phi ] = 0\), \([\partial _n {\widetilde{V}}\phi ] = -\phi \) for \(\widetilde{V}\) and additionally the jump relations \([\gamma _0 {\widetilde{K}}\phi ] = \phi \), \([\partial _n {\widetilde{K}}\phi ] = 0\) satisfied by \(\widetilde{K}\) from [22, Thm. 3.3.1], we observe that the function \(u:= \widetilde{{\mathcal {C}}}_{\eta }\varphi \) solves
The decay of u - the dominant part is the single-layer potential - and the Newton potential \({\mathcal {N}}(2\nabla \eta \cdot \nabla {\widetilde{K}}\varphi + (\varDelta \eta ) {\widetilde{K}}\varphi )\) for \(\left| x \right| \rightarrow \infty \) are the same, which allows us to write \(u={\mathcal {N}}(2\nabla \eta \cdot \nabla {\widetilde{K}}\varphi + (\varDelta \eta ) {\widetilde{K}}\varphi )\). With the mapping properties of the Newton potential and the standard mapping properties of \({\widetilde{K}}\) from [22, Thm. 3.1.16], we get
The trace estimate applied facewise as in the proof of Lemma 3.1, and estimates (3.3), (3.22) lead to
Similarly, we obtain with Lemma 3.2, (i)
Next, we identify \(\partial _n \widetilde{\mathcal C}_\eta \). With \(W = -\partial _n \widetilde{K}\), 
, and 
, we compute
Recalling the mapping properties \(K',K:H^s(\varGamma )\rightarrow H^s(\varGamma )\) and the relation 
, we get with the aid of (3.23), (3.24)
2. step: Since \(H^{1/2}(\varGamma )\) is dense in \(H^s(\varGamma )\), \(s \in (0,1/2)\), the operator \({\mathcal C}_\eta \) can be extended (in a unique way) to a bounded linear operator \(H^s(\varGamma ) \rightarrow H^s(\varGamma )\).
3. step: The operator \({\mathcal C}_\eta \) is skew-symmetric: The operator W maps \(H^{1/2}(\varGamma ) \rightarrow H^{-1/2}(\varGamma )\) and is symmetric. The skew-symmetry of \({\mathcal C}_\eta \) then follows from a direct calculation.
4. step: The skew-symmetry of \({\mathcal C}_\eta \) allows us to extend (in a unique way) the operator as an operator \(H^{-s}(\varGamma ) \rightarrow H^{-s}(\varGamma )\) for \(0<s<1/2\) by the following argument: For \(\varphi \), \(\psi \in H^{1/2}(\varGamma )\) we compute
Since \({\mathcal C}_\eta : H^{s}(\varGamma ) \rightarrow H^{s}(\varGamma )\) for \(0< s < 1/2\), we see that \(\varphi \mapsto \langle \varphi , {\mathcal C}_\eta \psi \rangle \) of the right-hand side of (3.26) extends to a bounded linear functional on \(H^{-s}(\varGamma )\). Hence, \({\mathcal C}_\eta : H^{-s}(\varGamma ) \rightarrow H^{-s}(\varGamma )\) for \(0< s < 1/2\).
5. step: We have \({\mathcal C}_\eta : H^s(\varGamma ) \rightarrow H^s(\varGamma )\) for \(s \in (-1/2,1/2){\setminus } \{0\}\). An interpolation argument allows us to extend the boundedness to the remaining case \(s = 0\).
Proof of (ii): Let \(\widehat{\varphi } \in H^{-\alpha _N}(\varGamma )\). The argument leading to the first inequality in (3.22), i.e., the mapping properties of \(\mathcal {N}\), also shows
Since
the function \(v:=\widetilde{{\mathcal {C}}}_{\eta }^{\eta }\widehat{\varphi } {:}{=} \widetilde{{\mathcal {C}}}_{\eta }(\eta \widehat{\varphi })-\eta \widetilde{{\mathcal {C}}}_{\eta }\widehat{\varphi }\) solves
Again, the decay of v and the Newton potential applied to the right-hand side of the equation are the same. Hence, the mapping properties of the Newton potential provide
We apply Lemma 3.4 to \({\widetilde{K}}\widehat{\varphi }-\overline{{\widetilde{K}}\widehat{\varphi }},\) with \(\overline{{\widetilde{K}}\widehat{\varphi }} {:}{=} \frac{1}{\left| \varOmega \right| }\left\langle {\widetilde{K}}\widehat{\varphi },1 \right\rangle _{L^2(\varOmega )}\). Since we assumed 
, we have that \({\widetilde{K}}\widehat{\varphi }\) is smooth on \(\partial B_{R_{\varOmega }}(0)\), and we can estimate this term by an arbitrary negative norm of \(\widehat{\varphi }\) on \(\varGamma \) to obtain
The mean value \(\overline{\widetilde{K}\widehat{\varphi }}\) can be estimated with \(r^2 = \left| x \right| ^2\), the observation \(\varDelta r^2 = 2d\), and integration by parts by
where the last step follows since \(K'\) is a bounded operator mapping \(H^{\alpha _N}(\varGamma )\rightarrow H^{\alpha _N}(\varGamma )\) by Lemma 3.2. Using the mapping properties of W of Lemma 3.5, (iii) and inserting (3.29) in (3.28) leads together with a facewise trace estimate to
The computation
the mapping properties of V and the commutator of \(K'\) (as normal trace of the commutator \(\widetilde{{C}}_{\eta }\) from Lemma 3.3, cf. (3.9)) prove the lemma. \(\square \)
4 Proof of main results
With the consequences of the shift theorems from the previous section, we can prove our main results, the local error estimates for Symm’s integral equation and the hyper-singular integral equation.
4.1 Symm’s integral equation (proof of Theorem 2.3)
The main tools in our proofs are the Galerkin orthogonality
and a Caccioppoli-type estimate for discrete harmonic functions that satisfy the orthogonality
More precisely, we employ the space of discrete harmonic functions on an open set \(D\subset {\mathbb {R}}^d\) defined by
Proposition 4.1
[11, Lemma 3.9] For discrete harmonic functions \(u \in {\mathcal H}_{h}(B')\), the interior regularity estimate
holds, where B and \(B'\) are nested boxes and 
satisfies \(8h\le {\widehat{d}}\). The hidden constant depends only on \(\varOmega , d\), and the \(\gamma \)-shape regularity of \(\mathcal {T}_h\).
As a consequence of this interior regularity estimate and Lemma 3.1, we get an estimate for the jump of the normal derivative of a discrete harmonic single-layer potential.
Lemma 4.2
Let Assumption 2.1 hold and \(B \subset B' \subset B_{R_{\varOmega }}(0)\) be nested boxes with 
. Let h be sufficiently small so that the assumption of Proposition 4.1 holds. Let \(u{:}{=}{\widetilde{V}}\zeta _h\) with \(\zeta _h \in S^{0,0}(\mathcal {T}_h)\) and assume \(u \in {\mathcal H}_h(B')\). Let \({\widehat{\varGamma }}\subset B\cap \varGamma \) and \(\eta \in C^{\infty }_0({\mathbb {R}}^d)\) be an arbitrary cut-off function satisfying \(\eta \equiv 1\) on \(B'\cap \varGamma \). Then,
The constant \(C>0\) depends only on \(\varOmega ,d,{\widehat{d}}\), the \(\gamma \)-shape regularity of \(\mathcal {T}_h\), \(\Vert \eta \Vert _{W^{1,\infty }({\mathbb {R}}^d)}\), and the constants appearing in Assumption 2.1.
Proof:
We split the function \(u = u_{\text {far}} + u_{\text {near}} \), where the near field \(u_{\text {near}}\) and the far field \(u_{\text {far}}\) solve the Dirichlet problems
We first consider \(\gamma _1^\mathrm{int} u_\mathrm{near}\) - the case \(\gamma _1^\mathrm{ext} u_\mathrm{near}\) is treated analogously.
Let \({\widehat{\eta }}\) be another cut-off function satisfying \({\widehat{\eta }} \equiv 1\) on \({\widehat{\varGamma }}\) and 
. The multiplicative trace inequality, see, e.g., [16, Thm. A.2], implies for any \(\varepsilon \le 1/2\) that
Since \(u_{\text {near}} \in {\mathcal H}_h(B')\), we use the interior regularity estimate (4.2) for the first term on the right-hand side of (4.6), and the second term of (4.6) can be estimated using (3.2). In total, we get for \(\varepsilon \le \alpha _D<1/2\) that
Let \(\mathcal {I}_h:C(\varGamma ) \rightarrow S^{1,1}({{\mathcal {T}}}_h)\) be the nodal interpolation operator. The mapping properties of V from Lemma 3.2, (ii), the commutator \(C_{\eta }\) from (3.5) as well as an inverse inequality, see, e.g., [13, Thm. 3.2, Thm. 3.6], lead to
With the classical a priori estimate for the inhomogeneous Dirichlet problem in the \(H^1\)-norm, the commutator \(C_{\eta }\), and Lemma 3.3, we estimate
We apply (3.1), (since \(\eta \equiv 0\) on \(\partial B_{R_{\varOmega }}(0)\) only the boundary terms for \(\varGamma \) appear) together with Young’s inequality \(ab \le a^p/p + b^q/q\) applied with \(p=(1+2\varepsilon )/2\varepsilon \), \(q=1+2\varepsilon \) to obtain
Similarly, we get for the second term in (4.7)
Inserting everything in (4.7) and using \(h\lesssim 1\) gives
Applying the same argument for the exterior Dirichlet boundary value problem leads to an estimate for the jump of the normal derivative
It remains to estimate the far field \(u_{\text {far}}\), which can be treated similarly to the near field using a trace estimate and Lemma 3.1. Applying Lemma 3.1 with a cut-off function \({\widetilde{\eta }}\) satisfying \({\widetilde{\eta }} \equiv 1\) on \(B\cap \varGamma \) and 
, the boundary term in (3.2) disappears since \({\widetilde{\eta }}(1-\eta )\equiv 0\), which simplifies the arguments:
which proves the lemma. \(\square \)
We use the Galerkin projection \(\varPi _V: H^{-1/2}(\varGamma ) \rightarrow S^{0,0}(\mathcal {T}_h)\), which is defined by
We denote by \(I_h\) the \(L^2(\varGamma )\)-orthogonal projection given by
The operator \(I_h\) has the following super-approximation property, [18]: For any discrete function \(\psi _h \in S^{0,0}(\mathcal {T}_h)\) and a cut-off function \(\eta \), we have (with implied constants depending on \(\Vert \eta \Vert _{W^{1,\infty }}\))
The following lemma provides an estimate for the local Galerkin error and includes the key steps to the proof of Theorem 2.3.
Lemma 4.3
Let the assumptions of Theorem 2.3 hold. Let \(\widehat{\varGamma _0}\) be an open subset of \(\varGamma \) with \(\varGamma _0\subset \widehat{\varGamma _0} \subsetneq \varGamma \) and 
. Let h satisfy \(\frac{h}{R}\le \frac{1}{12}\). Assume that \(\phi \in L^{2}(\widehat{\varGamma _0})\). Then, we have for the Galerkin error \(\phi - \phi _h = \phi - \varPi _V\phi \)
The constant \(C>0\) depends only on \(\varGamma ,\varGamma _0,d,R,\) and the \(\gamma \)-shape regularity of \(\mathcal {T}_h\).
Proof:
We define \(e{:}{=} \phi -\phi _h\), open subsets \(\varGamma _0\subset \varGamma _1\subset \varGamma _2 \dots \subset \varGamma _5 \subset \widehat{\varGamma _0}\), and volume boxes \(B_0 \subset B_1 \subset B_2 \dots \subset B_5 \subset {\mathbb {R}}^d\), where \(B_i\cap \widehat{\varGamma }_0 = \varGamma _i\). Throughout the proof, we use cut-off functions \(\eta _i \in C_0^{\infty }({\mathbb {R}}^d)\), \(i=1,\dots ,5\), satisfying \(\eta _i \equiv 1\) on \(\varGamma _{i-1}\), 
and \(\left\| \nabla \eta _i \right\| _{L^{\infty }(B_i)}\lesssim \frac{1}{R}\). We write
With the Galerkin projection \(\varPi _V\) from (4.11), we obtain
With an inverse inequality and the \(L^2\)-orthogonal projection \(I_h\), which satisfies the super-approximation property (4.12) for \(\eta _5 \phi _h\), we get
where the last estimate follows from Céa’s lemma and super-approximation. The same argument leads to
In fact, this argument shows \(L^2\)-stability of \(\varPi _V\):
The bounds (4.15), (4.16) together imply
For the first term on the right-hand side of (4.14), we want to use Lemma 4.2. Since \([\partial _n{\widetilde{V}}\zeta _h] = -\zeta _h \in S^{0,0}(\mathcal {T}_h)\) for any discrete function \(\zeta _h \in S^{0,0}(\mathcal {T}_h)\), we need to construct a discrete function satisfying the orthogonality condition (4.2). Using the Galerkin orthogonality with test functions \(\psi _h\) with support 
and noting that \(\eta _5 \equiv 1\) on 
, we obtain with the commutator \(C_{\eta _5}\) defined in (3.5)
Thus, defining
we get on the volume box \(B_4\subset {\mathbb {R}}^d\) a discrete harmonic function
The correction \(\xi _h\) can be estimated using the \(L^2\)-stability (4.17) of the Galerkin projection, the mapping properties of \(V^{-1}\), \(C_{\eta _5}\), \(C_{\eta _5}^{\eta _5}\) from Lemma 3.3 by
We write
For the second term in (4.22) we estimate
We treat the first term in (4.22) as follows: We apply Lemma 4.2 with the boxes \(B_2\) and \(B_3\) (note that, since we assumed \(12 h\le R\), the condition 
can be fulfilled) to the discrete harmonic function \(u :={\widetilde{V}}\zeta _h \in {\mathcal H}_h(B_4)\) and the cut-off function \(\eta _4\). The jump condition \([\partial _n u] = -\zeta _h\) leads to
The definition of \(\zeta _h\), the bound (4.21), and the \(H^{-1/2}\)-stability of the Galerkin projection lead to
With the \(L^2\)-stability (4.17) of the Galerkin projection and (4.21) we get
We use the orthogonality (4.19) satisfied by \(\zeta _h\) on \(\varGamma _4\), the \(L^2\)-orthogonal projection \(I_h\) and the properties of the commutator \(C_{\eta _5}\) given by Lemma 3.3 to arrive at
Inserting (4.25)–(4.27) in (4.24) and using \(h\lesssim 1\), we arrive at
Combining (4.14), (4.22) with (4.18), (4.23), (4.28), and finally (4.13), we get
Since we only used the Galerkin orthogonality as a property of the error \(\phi -\phi _h\), we may write \(\phi -\phi _h = (\phi -\chi _h)+(\chi _h - \phi _h)\) for arbitrary \(\chi _h \in S^{0,0}(\mathcal {T}_h)\), and we have proven the inequality claimed in Lemma 4.3. \(\square \)
In order to prove Theorem 2.3, we need a lemma:
Lemma 4.4
For every \(\delta > 0\) there is a bounded linear operator \(J_\delta :H^{-1}(\varGamma ) \rightarrow L^{2}(\varGamma )\) with the following properties:
-
(i)
(stability): For every \(-1 \le s \le t \le 0\) there is \(C_{s,t} > 0\) (depending only on s, t, \(\varOmega \)) such that \(\Vert J_\delta u\Vert _{H^{t}(\varGamma )} \le \delta ^{s-t} C_s\Vert J_\delta u\Vert _{H^s(\varGamma )} \) for all \(u \in H^s(\varGamma )\).
-
(ii)
(locality): for \(\omega \subset \varGamma \) the restriction \((J_\delta u)|_\omega \) depends only on \(u|_{\omega _{\delta }}\) with \(\omega _\delta {:}{=} \cup _{x \in \omega } B_\delta (x) \cap \varGamma \).
-
(iii)
(approximation): For every \(-1 \le t \le s \le 1\) there is \(C_{s,t} > 0\) (depending only on s, t, \(\varOmega \)) such that \(\Vert u - J_\delta u\Vert _{H^{t}(\varGamma )} \le C_{s,t} \delta ^{s-t} \Vert u\Vert _{H^s(\varGamma )}\) for all \(u \in H^s(\varGamma )\).
Proof:
Operators with such properties are obtained by the usual mollification procedure (on a length scale \(O(\delta )\) for domains in \({\mathbb {R}}^d\)). This technique can be generalized to the present setting of surfaces with the aid of localization and charts. We also mention [1, 7] where similar operators mapping into \(S^{1,1}({{\mathcal {T}}}_h)\) are constructed. \(\square \)
We are in position to prove our main result, a local estimate for the Galerkin-boundary element error for Symm’s integral equation in the \(L^2\)-norm.
Proof
(of Theorem 2.3): Starting with Lemma 4.3, it remains to estimate the two terms \(h^{\alpha _D/(1+2\alpha _D)}\left\| e \right\| _{L^{2}(\widehat{\varGamma _0})}\) and \(\left\| e \right\| _{H^{-1/2}(\widehat{\varGamma _0})}\), where \(e{:}{=} \phi -\phi _h\).
We start with the latter. Let \({\widehat{\eta }} \in C^{\infty }({\mathbb {R}}^d)\) be a cut-off-function with \({\widehat{\eta }} \equiv 1\) on \(\widehat{\varGamma _0}\), 
and \(\left\| \nabla {\widehat{\eta }} \right\| _{L^{\infty }}\lesssim \frac{1}{R}\). Let \({\widetilde{\eta }}\) be another cut-off function with \({\widetilde{\eta }}=1\) on \(B^{\widehat{\varGamma _0}}_{R/2+h}\cap \varGamma \) and 
, where 
. Select \(\delta = ch \) with a constant \(c = O(1)\) such that the operator \(J_{ch}\) of Lemma 4.4 has the support property 
. We will employ the operator \(I_h \circ J_{ch}:H^{-1}(\varGamma ) \rightarrow S^{0,0}(\varGamma )\) with the \(L^2\)-orthogonal projection \(I_h\). It is easy to see that we may assume that
Concerning the approximation properties, we have
With the definition of the commutators \(C_{{\widehat{\eta }}}\), \(C_{\widehat{\eta }}^{{\widehat{\eta }}}\), the Galerkin orthogonality satisfied by e, and the fact that \(V:H^{-1/2}(\varGamma ) \rightarrow H^{1/2}(\varGamma )\) is an isomorphism, we get
The first term on the right-hand side of (4.31) can be treated in the same way as the term \(h^{\alpha _D/(1+2\alpha _D)}\left\| e \right\| _{L^{2}(\widehat{\varGamma _1})}\) on the right-hand side of Lemma 4.3, which is treated by iterating the \(L^2\)-estimate of the statement of Theorem 4.3. That is, we set \(m{:}{=}\lceil \frac{(1+\alpha _D)(1+2\alpha _D)}{\alpha _D} \rceil \). The assumption \(C_{\alpha _D}\frac{h}{R}\le \frac{1}{12}\) allows us to define m nested domains \(\widehat{\varGamma _i}\), \(i=0,\dots ,m-1\) such that 
, \(\widehat{\varGamma }_m\subset {\widehat{\varGamma }}\). Since the term \(h^{\alpha _D/(1+2\alpha _D)}\left\| e \right\| _{L^{2}(\widehat{\varGamma _1})}\) again contains a local \(L^2\)-norm, we may use Lemma 4.3 and (4.31) again on the larger set \(\widehat{\varGamma _2}\subsetneq \varGamma \) to estimate
Inserting this in the initial estimate of Lemma 4.3 (using \(h\lesssim 1\)) leads to
Now, the \(L^2\)-term on the right-hand side is multiplied by \(h^{2\alpha _D/(1+2\alpha _D)}\), i.e., the square of the initial factor. Iterating this argument \(m-2\)-times, provides the factor \(h^{m\alpha _D/(1+2\alpha _D)}\), and by the choice of m, we have \(h^{1+\alpha _D}\le h^{m\alpha _D/(1+2\alpha _D)}\). Together with an inverse estimate, we obtain
which proves the theorem. \(\square \)
Proof
(of Corollary 2.4): The assumption \(\phi \in H^{-1/2+\alpha }(\varGamma ) \cap H^{\beta }(\widetilde{\varGamma })\) leads to
where the second estimate is the standard global error estimate for the BEM, see [22].
It remains to estimate \(\left\| e \right\| _{H^{-1-\alpha _D}(\varGamma )}\), which is treated with a duality argument: We note that Assumption 2.1 and the jump relations imply the following shift theorem for V: If \(w \in H^{1+\alpha _D}(\varGamma )\) and \(\psi \) solves \(V\psi = w \in H^{1+\alpha _D}(\varGamma )\), then \(\psi \in H^{\alpha _D}(\varGamma )\) and \(\Vert \psi \Vert _{H^{\alpha _D}(\varGamma )} \lesssim \Vert w\Vert _{H^{1+\alpha _D}(\varGamma )}\). Hence, with the Galerkin projection \(\varPi _V\), we estimate
Therefore, the term of slowest convergence is of order \(\mathcal {O}(h^{\min \{1/2+\alpha +\alpha _D,\beta \}})\), which proves the corollary. \(\square \)
Remark 4.5
The term of slowest convergence in the case of high local regularity is the global error in the negative \(H^{-1-\alpha _D}(\varGamma )\)-norm, which is treated with a duality argument that uses the maximum amount of additional regularity on the polygonal/polyhedral domain. Therefore, further improvements of the convergence rate cannot be achieved with our method of proof. In fact, the numerical examples in the next section confirm the sharpness of this observation, i.e., that the best possible convergence is \(\mathcal {O}(h^{1/2+\alpha +\alpha _D})\).
The trivial estimate \(\left\| \eta e \right\| _{H^{-1/2}(\varGamma )} \lesssim \left\| \eta e \right\| _{L^{2}(\varGamma )} \) immediately implies that the local convergence in the energy norm is at least of order \(\mathcal {O}(h^{1/2+\alpha +\alpha _D})\) as well. Again, analyzing the proof of Lemma 4.3, we observe that an improvement is impossible, since the limiting term is once more the error in the negative \(H^{-1-\alpha _D}(\varGamma )\)-norm. \(\square \)
Remark 4.6
Remark 4.5 states that the local rate of convergence is limited by the shift theorem of Assumption 2.1. If the geometry \(\varOmega \) is smooth, then elliptic shift theorems for the Dirichlet problem hold in a wider range, e.g., if \(f\in H^{1/2}(\varOmega )\), we may get \(u \in H^{5/2}(\varOmega )\). It can be checked that in this setting, an estimate of the form
is possible since the commutator \(C_{\eta _5}^{\eta _5}\) in (4.21) then maps \(H^{-2}(\varGamma )\rightarrow H^1(\varGamma )\). If an even better shift theorem holds, then the \(H^{-2}\)-norm can be further weakened by using commutators of higher order. The best possible achievable local rates are then \(O(h^\beta )\) in \(L^2(\varGamma _0)\) for \(\phi \in H^\beta ({\widehat{\varGamma }})\), \(\beta \in [0,1]\) and \(O(h^{1/2+\beta })\) in the \(H^{-1/2}(\varGamma _0)\)-norm.
\(\square \)
4.2 The hyper-singular integral equation (proof of Theorem 2.8)
We start with the Galerkin orthogonality
and a Caccioppoli-type estimate on \(D\subset {\mathbb {R}}^d\) for functions characterized by the orthogonality
for some \(\mu \in {\mathbb {R}}\). Here, we define the space of discrete harmonic functions \({\mathcal H}^{{\mathcal {N}}}_{h}(D,\mu )\) for an open set \(D\subset {\mathbb {R}}^d\) and \(\mu \in {\mathbb {R}}\) as
Proposition 4.7
[12, Lemma 3.8] For discrete harmonic functions \(u \in {\mathcal H}^{{\mathcal {N}}}_{h}(B',\mu )\), we have the interior regularity estimate
where B and \(B'\) are nested boxes and 
satisfies \(8h\le {\widehat{d}}\). The hidden constant depends only on \(\varOmega , d\), and the \(\gamma \)-shape regularity of \(\mathcal {T}_h\).
We use the Galerkin projection \(\varPi _W: H^{1/2}(\varGamma ) \rightarrow S^{1,1}(\mathcal {T}_h)\), now defined by
The following lemma collects approximation properties of the Galerkin projection that will be applied in both Lemmas 4.10 and 4.11 below.
Lemma 4.8
Let \(\varPi _W\) be the Galerkin projection defined in (4.36), and let \(\eta \), \(\widehat{\eta } \in C_0^{\infty }(\mathbb {R}^d)\) be with \(\widehat{\eta } \equiv 1\) on 
. For \(\varphi \in H^1(\varGamma )\), we have for \(s \in [1/2,1]\)
For \(\varphi _h \in S^{1,1}(\mathcal {T}_h)\), we have for \(s \in [1/2,1]\)
The constant \(C>0\) depends only on \(\varOmega \), the \(\gamma \)-shape regularity of \(\mathcal {T}_h\), and \(\Vert \eta \Vert _{W^{2,\infty }(\mathbb {R}^d)}\).
Proof:
Let \(\mathcal {J}_h\) be a quasi-interpolation operator with approximation properties in the \(H^s\)-seminorm, e.g., the Scott-Zhang-projection, [24]. We use super-approximation similarly to (4.12). Since \(\varphi _h \in S^{1,1}(\mathcal {T}_h)\), we have to use the piecewise \(H^2\)-norm, and an inverse inequality leads to
where, in the last step, the assumption on \(\widehat{\eta }\) was used. Similarly, the \(H^1\)-norm estimate \(\left\| \eta \varphi _h - \mathcal {J}_h(\eta \varphi _h) \right\| _{H^1(\varGamma )} \lesssim h^{s}\left\| \widehat{\eta }\varphi _h \right\| _{H^s(\varGamma )}\) holds. Interpolation finally leads to a super-approximation result in \(H^s\)
With an inverse inequality, see, e.g., [13, Thm. 3.2], as well as Céa’s lemma this implies
A similar argument leads to
and consequently to the \(H^1\)-stability of the Galerkin-projection. \(\square \)
In the following, we need stability and approximation properties of the Scott-Zhang projection \({\mathcal {J}}_h\) in the space \(H^{1+\alpha _N}(\varGamma )\) provided by the following lemma.
Lemma 4.9
Let \({\mathcal {J}}_h\) be the Scott-Zhang projection defined in [24]. Then, for \(s \in [0,3/2)\) we have
and therefore, for every \(0 \le t \le s < 3/2\)
The constants \(C_s\), \(C_{s,t}>0\) depend only on \(\varOmega \), the \(\gamma \)-shape regularity of \(\mathcal {T}_h\), and s, t.
Proof:
We start with the proof of (4.39). The stability for the case \(s = 1\) is given in [24] and the stability for the case \(s = 0\) (note that \(\varGamma \) is a closed surface without boundary) is discussed in [3, Lemma 7]. By interpolation, (4.39) follows for \(0< s < 1\). The starting point for the proof of (4.39) for \(s \in (1,3/2)\) is that, by Remark 1.1, (iii), we may focus on a single affine piece \(\varGamma _i\) of \(\varGamma \) and can exploit that the notion of \(H^s(\varGamma _i)\) coincides with the standard notion on intervals (in 1D) and polygons (in 2D). In particular, \(H^s(\varGamma _i)\) can be defined as the interpolation space between \(H^1(\varGamma _i)\) and \(H^2(\varGamma _i)\). Since \({{\mathcal {J}}}_h u \in C^0(\varGamma )\), Remark 1.1, (iii) implies for \(s \in (1,3/2)\)
It therefore suffices to show \(\Vert {{{\mathcal {J}}}}_h u\Vert _{H^s(\varGamma _i)} \le C \Vert u\Vert _{H^s(\varGamma _i)}\). Since \(H^s(\varGamma _i)\) is an interpolation space between \(H^1(\varGamma _i)\) and \(H^2(\varGamma _i)\), we can find (cf. [4]), for every \(t > 0\), a function \(u_t \in H^2(\varGamma _i)\) with
Let \(I_h'\) be an approximation operator with the simultaneous approximation property
see, e.g., [4, 6, Thm. 14.4.2]. With an inverse inequality, cf. [8, Appendix], the \(H^1\)-stability of the Scott-Zhang projection, and (4.41), (4.42), we estimate
Choosing \(t=\mathcal {O}(h)\), we get the \(H^{s}(\varGamma _i)\)-stability of \({\mathcal {J}}_h\) and thus also the \(H^s(\varGamma )\)-stability of \({{\mathcal {J}}}_h\).
We only prove the approximation property (4.40) for \(s \in (1,3/2)\) as the case \(s \in [0,1]\) is covered by standard properties of the Scott-Zhang operator.
Case \(1 \le t \le s <3/2\): we observe with the stability properties of \({{{\mathcal {J}}}}_h\) and the approximation properties of \(I_h'\)
Case \(t=0\): we observe with the stability properties of \({{{\mathcal {J}}}}_h\) and the approximation properties of \(I_h'\)
Case \(0< t < 1\): The remaining cases are obtained with the aid of an interpolation inequality:
which concludes the proof. \(\square \)
The following lemma is similar to Lemma 4.2. Here, we obtain an estimate for the jump of the trace of a discrete harmonic double-layer potential.
Lemma 4.10
Let Assumption 2.7 hold and let \(B \subset B' \subset B''\) be nested boxes with 
. Let h be sufficiently small so that the assumption of Proposition 4.7 holds. Let \(u{:}{=}{\widetilde{K}}\zeta _h\) with \(\zeta _h \in S^{1,1}(\mathcal {T}_h)\) and assume \(u \in {\mathcal H}^{{\mathcal {N}}}_h(B'',\mu )\) for the box \(B'' \subset B_{R_{\varOmega }}(0)\) and some \(\mu \in {\mathbb {R}}\). Let \({\widehat{\varGamma }}\subset B\cap \varGamma \). Then,
The constant \(C>0\) depends only on \(\varOmega ,{\widehat{d}}\), the \(\gamma \)-shape regularity of \(\mathcal {T}_h\), and the constants appearing in Assumption 2.7.
Proof:
Step 1 (Splitting into near and far-field): Let \(\eta \in C^{\infty }_0({\mathbb {R}}^d)\) satisfy \(\eta \equiv 1\) on \(B'\cap \varGamma \) and 
. Define the near-field \(u_{\text {near}}\) and the far field \(u_{\text {far}}\) as potentials \(u_{\text {near}}{:}{=} {\widetilde{K}}v_h - \overline{{\widetilde{K}}v_h}\) with \(\overline{{\widetilde{K}}v_h}{:}{=}\frac{1}{\left| \varOmega \right| }\int _{\varOmega }{{\widetilde{K}}v_h}\) and \(u_{\text {far}}{:}{=}{\widetilde{K}}\nu _h-\overline{{\widetilde{K}}\nu _h}\), where \(v_h, \nu _h \in S^{1,1}(\mathcal {T}_h)\) are BEM solutions of
with \(\left\langle v_h,1 \right\rangle =0=\left\langle \nu _h,1 \right\rangle \). Here, z is a function with \(z\equiv \mu \) on \(\varGamma \cap B'\) such that the compatibility condition \(\left\langle \eta W\zeta _h - \eta z,1 \right\rangle =\left\langle (\eta -1)W\zeta _h - \eta z,1 \right\rangle = 0\) holds. Since \(\left\langle W\zeta _h,1 \right\rangle =0\) such a function exists. More precisely, we choose \(z \in L^2(\varGamma )\) to be the piecewise constant function
The function \(v_h+\nu _h\) solves
which implies \(v_h+\nu _h = \zeta _h + c\) for a constant c. Therefore, \(v{:}{=}u_{\text {near}}+u_{\text {far}} = u + {\widetilde{K}}c - \overline{{\widetilde{K}}(v_h+\nu _h)}\). Since \([\gamma _0 {\widetilde{K}}c] = c\) this implies
The definition of z and \(\eta \equiv 1\) on \(B' \cap \varGamma \) lead to
Consequently, we obtain
The last inequality follows from the orthogonality of \(W\zeta _h\) to discrete functions in \(S^{1,1}(\mathcal {T}_h)\) on \(B''\) and the arguments shown in (4.47) below (specifically: go through the arguments of (4.47) with \(z \equiv \mu \)).
Step 2 (Approximation of the near field):
Let \({\mathcal {J}}_h\) denote the Scott–Zhang projection. The ellipticity of W on \(H^{1/2}(\varGamma )/{\mathbb {R}}\) and the orthogonality (4.33) of \(W\zeta _h = -\partial _n {\widetilde{K}}\zeta _h\) imply
With the same arguments and Lemma 4.9 we may estimate
Let \(\psi \) solve \(W\psi = w-\overline{w}\) for \(w \in H^{\alpha _N}(\varGamma )\). Then \(\psi \in H^{1+\alpha _N}(\varGamma )\). Together with the mapping properties of W from Lemma 3.5, \(\left\langle v_h,1 \right\rangle = 0\), the definition of \(v_h\), and the stability and approximation properties of \({\mathcal {J}}_h\) from Lemma 4.9, we obtain
With the mapping properties of W from Lemma 3.5, an inverse estimate, and (4.47) we obtain for \(0\le \varepsilon \le \alpha _N\)
We first consider \(\gamma _0^\mathrm{int} u_\mathrm{near}\); the case \(\gamma _0^\mathrm{ext} u_\mathrm{near}\) is treated analogously. By construction of \(u_\mathrm{near}\), we have
since \(z\equiv \mu \), \(\eta \equiv 1\) on 
. Therefore, \(u_{\text {near}} \in {\mathcal H}^{{\mathcal {N}}}_h(B',0)\).
Let \({\widehat{\eta }}\) be another cut-off function satisfying \({\widehat{\eta }} \equiv 1\) on \({\widehat{\varGamma }}\) and 
. The multiplicative trace inequality, see, e.g., [16, Thm. A.2], implies for any \(\varepsilon \le 1/2\) that
Since \(u_{\text {near}} \in {\mathcal H}^{{\mathcal {N}}}_h(B',0)\), we may use the interior regularity estimate (4.35) with \(\mu = 0\) for the first term on the right-hand side of (4.52). The second factor of (4.52) can be estimated using (3.16) of Lemma 3.4. In total, we get for \(\varepsilon \le \alpha _N<1/2\) that
The mapping properties of \({\widetilde{K}}\) imply with (4.47) and (4.50)
We apply (3.15) (note: \(u_{\text {near}}\) has mean zero) and since \({\widetilde{K}} v_h\) is smooth on \(\partial B_{R_{\varOmega }}(0)\), we can estimate \(\left\| {\widetilde{K}}v_h \right\| _{H^{-\alpha _N}(\partial B_{R_{\varOmega }}(0))} \lesssim \left\| v_h \right\| _{H^{-\alpha _N}(\varGamma )}\). Together with (4.50), (4.49), and Young’s inequality this leads to
Similarly, with (4.54) we get for the second term in (4.53)
Inserting everything in (4.53) and choosing \(\varepsilon = \alpha _N\) gives
Applying the same argument for the exterior trace leads to an estimate for the jump of the trace
Step 3 (Approximation of the far field):
We define the function \(\nu \in H^{1/2}(\varGamma )\) as the solution of
Then, we have
Let \(\widehat{u}_\mathrm{far} {:}{=} {\widetilde{K}}\nu -\overline{{\widetilde{K}}\nu }\) where \(\overline{{\widetilde{K}}\nu }{:}{=}\frac{1}{\left| \varOmega \right| }\left\langle {\widetilde{K}}\nu ,1 \right\rangle _{L^2(\varOmega )}\) and \({\widehat{\eta }}\) be another cut-off function with \({\widehat{\eta }} \equiv 1\) on \({\widehat{\varGamma }}\) and 
. Then, with the Galerkin projection \(\varPi _W\), the triangle inequality and the jump conditions of \({\widetilde{K}}\) imply
The smoothness of \({\widetilde{K}}\nu \) on \(\partial B_{R_{\varOmega }}(0)\) and the coercivity of W on \(H^{1/2}(\varGamma )/{\mathbb {R}}\) lead to
We apply Lemma 3.4 with a cut-off function \({\widetilde{\eta }}\) satisfying \({\widetilde{\eta }} \equiv 1\) on \(B\cap \varGamma \) and 
. Then \(\eta \equiv 1\) and \(z \equiv \mu \) on \(B'\cap \varGamma \) imply \({\widetilde{\eta }}(1-\eta )\equiv 0\) and \({\widetilde{\eta }} \eta z = {\widetilde{\eta }} \mu \). The \(H^1\)-stability of the Galerkin projection from Lemma 4.8, a facewise trace estimate, and similar estimates as for the near field imply
It remains to estimate the first term on the right-hand side of (4.55). With an inverse estimate and Lemma 4.8 we get
We use the abbreviation \(e_{\nu } {:}{=} \nu - \nu _h\). The ellipticity of W on \(H^{1/2}(\varGamma )/{\mathbb {R}}\) and the definition of the Galerkin projection \(\varPi _W\) imply
With the commutator \({\mathcal {C}}_{\widehat{\eta }}\) we get
The definition of the Galerkin projection and the super-approximation properties of the Scott-Zhang projection \({\mathcal {J}}_h\) lead to
For the term involving \(\mathcal {C}_{\widehat{\eta }}\) in (4.58), we get with Lemma 3.6
A duality argument implies \(\left\| e_{\nu } \right\| _{H^{-\alpha _N}(\varGamma )}\lesssim h^{1/2+\alpha _N}\left\| \nu \right\| _{H^{1/2}(\varGamma )}\), for details we refer to the proof of Corollary 2.9. Inserting everything in (4.57) leads to
Finally, this implies with (4.55) and (4.56) that
which proves the lemma. \(\square \)
Lemma 4.11
Let \(\varphi ,\varphi _h\) be solutions of (2.8), (2.9) and let \(\varGamma _0, {\widehat{\varGamma }}\) be subsets of \(\varGamma \) with \(\varGamma _0\subset {\widehat{\varGamma }} \subsetneq \varGamma \) and 
. Let h be such that \(\frac{h}{R}\le \frac{1}{12}\), and let \(\eta \in C_0^{\infty }({\mathbb {R}}^d)\) satisfy \(\eta \equiv 1\) on \(\varGamma _0\), 
. Then, we have
with a constant \(C>0\) depending only on \(\varGamma ,\varGamma _0,{\widehat{\varGamma }},d,R\), and the \(\gamma \)-shape regularity of \(\mathcal {T}_h\).
Proof:
We define \(e{:}{=} \varphi -\varphi _h\), subsets \(\varGamma _0\subset \varGamma _1\subset \varGamma _2 \subset \varGamma _3\subset \varGamma _4 \subset {\widehat{\varGamma }}\), and volume boxes \(B_0 \subset B_1 \subset B_2 \subset B_3 \subset B_4\subset {\mathbb {R}}^d\), where \(B_i\cap \widehat{\varGamma } = \varGamma _i\). Throughout the proof, we use cut-off functions \(\eta _i \in C_0^{\infty }({\mathbb {R}}^d)\), \(i=1,\dots ,4\). These smooth functions \(\eta _i\) should satisfy \(\eta _i \equiv 1\) on \(\varGamma _{i-1}\), 
.
We want to use Lemma 4.10. Since \([\gamma _0{\widetilde{K}}\zeta _h] = \zeta _h \in S^{1,1}(\mathcal {T}_h)\) for any discrete function \(\zeta _h \in S^{1,1}(\mathcal {T}_h)\), we need to construct a discrete function satisfying the orthogonality (4.33). Using the Galerkin orthogonality with test functions with 
and noting that \(\eta _3 \equiv 1\) on 
, we obtain with the commutator \({\mathcal {C}}_{\eta _3}\) defined in (3.18), the abbreviation \(\overline{\eta _3{\mathcal {C}}_{\eta _3}e}=\frac{1}{\left| \varGamma \right| }\left\langle \eta _3 {\mathcal {C}}_{\eta _3}e,1 \right\rangle \), and the Galerkin projection \(\varPi _W\) from (4.36)
Here and below, we understand the inverse \(W^{-1}\) as the inverse of the bijective operator \(W:H^{1/2}_{*}(\varGamma ){:}{=}\{v\in H^{1/2}(\varGamma ):\left\langle v,1 \right\rangle = 0\}\rightarrow H_{*}^{-1/2}(\varGamma ){:}{=}\{v\in H^{-1/2}(\varGamma ):\left\langle v,1 \right\rangle = 0\}\). Since \(W^{-1}\) maps into \(H_{*}^{1/2}(\varGamma )\) no additional terms in the orthogonality (4.59) appear. Thus, defining
we get on a volume box \(B_2\subset {\mathbb {R}}^d\) a discrete harmonic function
where \(\mu = \left\langle e,1 \right\rangle -\frac{1}{\left| \varGamma \right| }\left\langle \eta _3 {\mathcal {C}}_{\eta _3}e,1 \right\rangle -\left\langle \eta _3e -\varPi _W(\eta _3e),1 \right\rangle \).
With the Galerkin projection \(\varPi _W\) from (4.36) and \(\eta _3 \equiv 1\) on 
, we write
Lemma 4.8 leads to
Using the \(H^1\)-stability of the Galerkin projection \(\varPi _W\), the mapping properties of \(W^{-1}\) and \({\mathcal {C}}_{\eta _3}\) as well as Lemma 3.6, the correction \(\xi _h\) can be estimated by
For the second term on the right-hand side of (4.60) we have \(\left\| \eta _1\zeta _h \right\| _{H^{1}(\varGamma )}\lesssim \left\| \eta _1\nabla \zeta _h \right\| _{L^{2}(\varGamma )} + \left\| \zeta _h \right\| _{L^{2}(\varGamma )}\). We apply Lemma 4.10 to \(u = {\widetilde{K}}\zeta _h \in \mathcal {H}_h^{{\mathcal {N}}}(B_2,\mu )\) and obtain
The \(H^1\)-stability of the Galerkin-projection from Lemma 4.8 and (4.62) lead to
as well as
With the estimate \(\left| \left\langle e,1 \right\rangle \right| \lesssim \left\| e \right\| _{H^{-\alpha _N}(\varGamma )}\) and previous arguments (using (4.62), Lemma 4.8, and Lemma 3.6), we get
Inserting (4.64)–(4.66) in (4.63), we arrive at
Combining (4.61), (4.62), and (4.67) in (4.60), we finally obtain
Since we only used the Galerkin orthogonality as a property of the error e, we may write \(\varphi -\varphi _h = (\varphi -\chi _h)+(\chi _h - \varphi _h)\) for arbitrary \(\chi _h \in S^{1,1}(\mathcal {T}_h)\) with 
and we have proven the claimed inequality. \(\square \)
Proof
(of Theorem 2.8): Starting from Lemma 4.11, it remains to estimate the terms \(h^{\alpha _N}\left| \varphi -\varphi _h \right| _{H^{1}({\widehat{\varGamma }})}\) and \(\left\| \eta (\varphi -\varphi _h) \right\| _{H^{1/2}({\widehat{\varGamma }})}\). The terms are treated as in the proof of Theorem 2.3. Rather than using the operator \(I_h\circ J_{ch}\) we may use the Scott-Zhang projection. \(\square \)
Proof
(of Corollary 2.9): The assumption \(\varphi \in H^{1/2+\alpha }(\varGamma ) \cap H^{1+\beta }(\widetilde{\varGamma })\) leads to
where the second estimate is the standard global error estimate for the Galerkin BEM applied to the hyper-singular integral equation, see [22].
For the remaining term in Theorem 2.8, we use a duality argument. Let \(\psi \) solve \(W\psi = w- \overline{w} \in H^{\alpha _N}(\varGamma )\), \(\left\langle \psi ,1 \right\rangle = 0\), where \(\overline{w} = \frac{1}{\left| \varGamma \right| }\left\langle w,1 \right\rangle \). Then \(\psi \in H^{1+\alpha _N}(\varGamma )\), and since \(\left\langle e,1 \right\rangle = 0\), we get with the Scott-Zhang projection \({\mathcal {J}}_h\) and Lemma 4.9
Therefore, the term of slowest convergence is of order \(\mathcal {O}(h^{\min \{1/2+\alpha +\alpha _N,\beta \}})\), which proves the corollary. \(\square \)
5 Numerical examples
In this section we provide numerical examples to illustrate the theoretical results of Sect. 2 and indicate their sharpness. We only consider Symm’s integral equation on quasi-uniform meshes. Provided the right-hand side and the geometry are sufficiently smooth, it is well-known that the lowest order boundary element method in two dimensions converges in the energy norm as \(O(N^{-3/2})\), where N denotes the degrees of freedom. In our examples we will consider problems, where the rate of convergence with uniform refinement is reduced due to singularities.
In order to compute the error between the exact solution and the Galerkin approximation, we prescribe the solution \(u(r,\theta ) = r^{\alpha }\cos (\alpha \theta )\) of Laplace’s equation in polar coordinates. Then, the normal derivative \(\phi = \partial _n u\) of u is the solution of
The regularity of \(\phi \) is determined by the choice of \(\alpha \). In fact, we have \(\phi \in H^{-1/2+\alpha -\varepsilon }(\varGamma )\), \(\varepsilon >0\), and locally \(\phi \in H^{1}(\widetilde{\varGamma })\) for all subsets \(\widetilde{\varGamma } \subset \varGamma \) that are a positive distance away from the singularity at the origin.
The lowest order Galerkin approximation to \(\phi \) is computed using the MATLAB-library HILBERT [2], where the errors in the \(L^2\)-norm are computed using two point Gauss-quadrature. The error in the local \(H^{-1/2}\)-norm is computed as \(\left\| \chi e \right\| _{H^{-1/2}(\varGamma )}^2\sim \left\langle V(\chi e),\chi e \right\rangle \), where \(\chi \) is the characteristic function for a union of elements \(\varGamma _0\subset \varGamma \).
L-shaped and Z-shaped domain, local error computed on fat, red part
5.1 Example 1: L-shaped domain
On the L-shaped domain depicted in Fig. 1 (left), the dual problem permits solutions of regularity \(H^{1/6-\varepsilon }(\varGamma )\) for arbitrary \(\varepsilon >0\); that is, we have \(\alpha _D = \frac{1}{6}-\varepsilon \).
Local and global convergence of Galerkin-BEM for Symm’s equation, L-shaped domain, left: \(\alpha = \frac{1}{3}\), right: \(\alpha = \frac{1}{8}\), \(\mathrm{loc} = \frac{19}{24}\)
Figure 2 shows the global convergence in the energy norm (blue squares) as well as the local convergence on the fat, red part of the boundary (\(\varGamma _0\), union of elements) in the \(L^2\)-norm (red stars) as well as the \(H^{-1/2}\)-norm (brown triangles). The black dotted lines mark the reference curves of order \(N^{-\beta }\) for various \(\beta > 0\).
In the left plot of Fig. 2 we chose \(\alpha = \frac{1}{3}\), which leads to \(\alpha + \alpha _D = \frac{1}{2}-\varepsilon \) and, indeed, we observe convergence in the local \(L^2\)-norm of almost order 1, which coincides with the theoretical rate obtained in Corollary 2.4. The error in the local \(H^{-1/2}\)-norm is smaller than the error in the \(L^2\)-norm, but does converge with the same rate, i.e., an improvement of Theorem 2.3 in the energy norm is not possible. The right plot in Fig. 2 shows the same quantities for the choice \(\alpha =\frac{1}{8}\). Obviously, in this case the rates of convergence are lower, and the local \(L^2\)-error does not converge with the best possible rate of one, but rather with the expected rate of \(N^{-19/24}=N^{-1/2-\alpha -\alpha _D}\), as predicted by Corollary 2.4.
Local and global convergence of Galerkin-BEM for Symm’s equation, Z-shaped domain, left: \(\alpha = \frac{1}{3}\), \(\mathrm{loc} = \frac{19}{21}\), right: \(\alpha = \frac{1}{8}\), \(\mathrm{loc} = \frac{39}{56}\)
5.2 Example 2: Z-shaped domain
We consider the Z-shaped geometry depicted in Fig. 1 (right). Here, the dual problem permits solutions of regularity \(H^{\alpha _D}(\varGamma )\) with \(\alpha _D = \frac{1}{14}-\varepsilon \). Again, we observe the expected convergence \(O(N^{-\alpha })\) for the global error in the energy norm in Fig. 3. However, in contrast to the previous example on the L-shaped domain, we do not obtain a rate of 1 for the local error in the \(L^2\)-norm for the case \(\alpha =\frac{1}{3}\), but rather a rate of \({-19/21}\), since \(\frac{1}{2}+\alpha _D + \alpha = \frac{19}{21}-\varepsilon \). For the choice \(\alpha =\frac{1}{8}\), we observe convergence \(\mathcal {O}(N^{-1/2-1/14-1/8}) = \mathcal {O}(N^{-39/56})\), which once more matches the theoretical convergence \(N^{-1/2-\alpha -\alpha _D}\).
References
Apel, T., Nicaise, S., Pfefferer, J.: Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains. Numer. Methods Partial Differ. Equ. 32(5), 1433–1454 (2016)
Aurada, M., Ebner, M., Feischl, M., Ferraz-Leite, S., Führer, T., Goldenits, P., Karkulik, M., Mayr, M., Praetorius, D.: HILBERT—a MATLAB implementation of adaptive 2D-BEM. Numer. Algorithms 67(1), 1–32 (2014)
Aurada, M., Feischl, M., Führer, T., Karkulik, M., Praetorius, D.: Energy norm based error estimators for adaptive BEM for hypersingular integral equations. Appl. Numer. Math. 95, 15–35 (2015)
Bramble, J., Scott, R.: Simultaneous approximation in scales of Banach spaces. Math. Comp. 32, 947–954 (1978)
Braun, K.: Interior error estimates of the Ritz method for pseudodifferential equations. Jpn. J. Appl. Math. 3(1), 59–72 (1986)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, vol. 15. Springer, New York (2002)
Carstensen, C., Verfürth, R.: Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36, 1571–1587 (1999)
Dahmen, W., Faermann, B., Graham, I.G., Hackbusch, W., Sauter, S.A.: Inverse inequalities on non-quasiuniform meshes and application to the mortar element method. Math. Comp. 73, 1107–1138 (2001)
Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)
Demlow, A., Guzmán, J., Schatz, A.H.: Local energy estimates for the finite element method on sharply varying grids. Math. Comput. 80(273), 1–9 (2011)
Faustmann, M., Melenk, J.M., Praetorius, D.: Existence of \({\cal{H}}\)-matrix approximants to the inverse of BEM matrices: the simple-layer operator. Math. Comput. 85, 119–152 (2016)
Faustmann, M., Melenk, J.M., Praetorius, D.: Existence of \({\cal{H}}\)-matrix approximants to the inverse of BEM matrices: the hyper-singular integral operator. IMA J. Numer. Anal. 37(3), 1211–1244 (2017)
Graham, I.G., Hackbusch, W., Sauter, S.A.: Finite elements on degenerate meshes: inverse-type inequalities and applications. IMA J. Numer. Anal. 25(2), 379–407 (2005)
Hartmann, T., Stephan, E.: Local error estimates and post processing for the Galerkin method on polygons. preprint, University of Hannover pp. 1–18 (1996)
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Melenk, J.M.: On Approximation in Meshless Methods. Frontiers of numerical analysis. Universitext, pp. 65–141. Springer, Berlin (2005)
Mikhailov, S.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Anal. Appl. 378(1), 324–342 (2011)
Nitsche, J.A., Schatz, A.H.: Interior estimates for Ritz–Galerkin methods. Math. Comput. 28, 937–958 (1974)
Rannacher, R., Wendland, W.L.: On the order of pointwise convergence of some boundary element methods. I. Operators of negative and zero order. RAIRO Modél. Math. Anal. Numér. 19(1), 65–87 (1985)
Rannacher, R., Wendland, W.L.: Pointwise convergence of some boundary element methods. II. RAIRO Modél. Math. Anal. Numér. 22(2), 343–362 (1988)
Saranen, J.: Local error estimates for some Petrov–Galerkin methods applied to strongly elliptic equations on curves. Math. Comput. 48(178), 485–502 (1987)
Sauter, S.A., Schwab, C.: Boundary Element Methods, Springer Series in Computational Mathematics, vol. 39. Springer, Berlin (2011)
Schulz, H., Steinbach, O.: A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem. Calcolo 37(2), 79–96 (2000)
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer, New York (2008)
Stephan, E.P., Tran, T.: Localization and post processing for the Galerkin boundary element method applied to three-dimensional screen problems. J. Integral Equ. Appl. 8(4), 457–481 (1996)
Tran, T.: The \(K\)-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates. Math. Comput. 64(210), 501–513 (1995)
Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59(3), 572–611 (1984)
Wahlbin, L.: Local behavior in finite element methods. In: P. Ciarlet, J. Lions (eds.) Handbook of numerical analysis. Volume II: Finite element methods (Part 1), pp. 353–522. North Holland (1991)
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Faustmann, M., Melenk, J.M. Local convergence of the boundary element method on polyhedral domains. Numer. Math. 140, 593–637 (2018). https://doi.org/10.1007/s00211-018-0975-1
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DOI: https://doi.org/10.1007/s00211-018-0975-1