Adaptive BEM for elliptic PDE systems, part I: abstract framework, for weakly-singular integral equations

In the present work, we consider weakly-singular integral equations arising from linear second-order elliptic PDE systems with constant coefficients, including, e.g. linear elasticity. We introduce a general framework for optimal convergence of adaptive Galerkin BEM. We identify certain abstract conditions for the underlying meshes, the corresponding mesh-refinement strategy, and the ansatz spaces that guarantee that the weighted-residual error estimator is reliable and converges at optimal algebraic rate if used within an adaptive algorithm. These conditions are satisfied, e.g. for discontinuous piecewise polynomials on simplicial meshes as well as certain ansatz spaces used for isogeometric analysis. Technical contributions include the localization of (non-local) fractional Sobolev norms and local inverse estimates for the (non-local) boundary integral operators associated to the PDE system.

1. Introduction 1.1.State of the art.For the Laplace model problem, adaptive boundary element methods (BEM) using (dis)continuous piecewise polynomials on triangulations have been intensively studied in the literature.In particular, optimal convergence of mesh-refining adaptive algorithms has been proved for polyhedral boundaries [FFK + 14, FFK + 15, FKMP13] as well as smooth boundaries [Gan13].The work [AFF + 17] allows to transfer these results to piecewise smooth boundaries; see also the discussion in the review article [CFPP14].In [BBHP19], these results have been generalized to the Helmholtz problem.In recent years, we have also shown optimal convergence of adaptive isogeometric BEM (IGABEM) using one-dimensional splines for the 2D Laplace problem [FGHP17,GPS20].However, the important case of 3D IGABEM remained open.Moreover, a generalization to other PDE operators is highly nontrivial (see (1.6) below), but especially linear elasticity is of great interest in the context of isogeometric analysis.
In [GHP17], we have considered isogeometric finite element methods (IGAFEM).We have derived an abstract framework which guarantees that, first, the classical residual FEM error estimator is reliable, and second, the related adaptive algorithm yields optimal convergence; see [GHP17, Section 2 and 4].We then showed that, besides standard FEM with piecewise polynomials, this abstract framework covers IGAFEM with hierarchical splines (see [GHP17, Section 3 and 5]) as well as IGAFEM with analysis suitable T-splines (see the recent work [GP19]).
The aim of the present work is to develop such an abstract framework also for BEM, which is mathematically much more demanding than FEM.In ongoing research [GP20], we aim to show that this framework covers, besides standard discretizations with piecewise polynomials, also IGABEM with hierarchical splines resp.T-splines.
To this end, the present work focusses on weakly-singular integral equations.For a given Lipschitz domain Ω ⊆ R d with compact boundary Γ := ∂Ω and right-hand side f : Γ → C, we consider (Vφ)(x) := Γ G(x − y)φ(y) dy = f (x) for almost all x ∈ Γ. (1.1) Here, the fundamental solution G stems from a strongly-elliptic PDE operator where the coefficients A ii ′ = A i ′ i ⊤ , b i , c ∈ C D×D are constant for some fixed dimension D ≥ 1.

Outline & Contributions.
In Section 2, we fix some general notation, recall Sobolev spaces on the boundary, and precisely state the considered problem.Section 3 can be paraphrased as follows: We formulate an adaptive algorithm (Algorithm 3.3) of the form driven by some weighted-residual a posteriori error estimator (see (1.4) below) in the frame of conforming Galerkin BEM.The algorithm particularly generates meshes T ℓ , BEM solutions Φ ℓ in associated nested ansatz spaces X ℓ ⊆ X ℓ+1 ⊂ L 2 (Γ) D ⊂ H −1/2 (Γ) D , and error estimators η ℓ for all ℓ ∈ N 0 .We formulate five assumptions (M1)-(M5) on the underlying meshes (Section 3.1), five assumptions (R1)-(R5) on the mesh-refinement (Section 3.2), and six assumptions (S1)-(S6) on the BEM spaces (Section 3.3).First, these assumptions are sufficient to guarantee that the a posteriori error estimator η ℓ associated with the BEM solution Φ ℓ is reliable, i.e., there exists a constant C rel > 0 such that where h ℓ ∈ L ∞ (Γ) denotes the local mesh-size function and ∇ Γ is the surface gradient.Second, Theorem 3.4 states that Algorithm 3.3 leads to linear convergence at optimal algebraic rate with respect to the number of mesh elements.In Theorem 3.8, we briefly note that the introduced conditions have already been implicitly proved for standard discretizations with piecewise polynomials on conforming triangulations.Moreover, we mention expected applications to adaptive IGABEM on quadrilateral meshes in Remark 3.9.Section 4 is devoted to the proof of Theorem 3.4.To prove reliability (1.4), we use a localization argument (Proposition 4.2), which generalizes earlier works [Fae00,Fae02] for standard discretizations.More precisely, we prove that |v| 2 H 1/2 (T ∪T ′ ) (1.5) for all v ∈ H1/2 (Γ) D that are L 2 -orthogonal onto the ansatz space X ℓ corresponding to some mesh T ℓ , where C split > 0 is independent of v and Π ℓ (T ) denotes the patch of T ∈ T ℓ .In Remark 4.10, we note that one obtains at least plain convergence lim ℓ→∞ φ−Φ ℓ H −1/2 (Γ) = 0 if Algorithm 3.3 is steered by the so-called Faermann estimator which is reliable and efficient.To prove linear convergence at optimal rate for the weightedresidual estimator (1.4), we show that the assumptions of Section 3 imply the axioms of adaptivity [CFPP14].The latter are properties for abstract mesh-refinements and abstract error estimators, which automatically yield the desired convergence result.In contrast to [FKMP13, FFK + 14] which (implicitly) verify the axioms of adaptivity only for the Laplace problem, our analysis allows for general PDE operators (1.2).The crucial step is the generalization (Proposition 4.13) of the non-trivial local inverse inequality for the non-local boundary integral operator V: With the help of a Caccioppoli-type inequality (Lemma 4.12), we prove that there exists a constant C inv > 0 such that (1.6) see [AFF + 17] for standard BEM discretizations of the Laplacian.Similar estimates hold also for the other boundary integral operators related to (1.2), namely the double-layer integral operator K, its adjoint K ′ , and the hypersingular integral operator W.These are stated and proved in Appendix B; again we refer to [AFF + 17] for standard BEM discretizations of the Laplacian.
While the present work focusses on the numerical analysis aspects only, we refer to the literature (see, e.g., [CMS01, ME14, FGHP16, Gan17, BBHP19]) for numerical experiments for the Laplace problem, the Helmholtz problem, and linear elasticity.

Preliminaries
In this section, we fix some general notation, recall Sobolev spaces on the boundary, and precisely state the considered problem.Throughout the work, let Ω ⊂ R d for d ≥ 2 be a bounded Lipschitz domain as in [McL00,Definition 3.28] and Γ := ∂Ω its boundary.
2.1.General notation.Throughout and without any ambiguity, |•| denotes the absolute value of scalars, the Euclidean norm of vectors in R n , as well as the d-dimensional measure of a set in We write A B to abbreviate A ≤ CB with some generic constant C > 0, which is clear from the context.Moreover, A ≃ B abbreviates A B A. Throughout, mesh-related quantities have the same index, e.g., X • is the ansatz space corresponding to the mesh T • .The analogous notation is used for meshes T • , T ⋆ , T ℓ etc.
. If σ > 0, and ω ⊆ Γ is an arbitrary measurable set, we define v H σ (ω) and |v| H σ (ω) analogously.With the definition . Note that H −σ (Γ) D with σ ∈ (0, 1] can be identified with the dual space of H σ (Γ) D , where we set As before, we abbreviate (2.10) The spaces H σ (Γ) can also be defined as trace spaces or via interpolation, where the resulting norms are always equivalent with constants depending only on the dimension d and the boundary Γ.More details and proofs are found, e.g., in the monographs [McL00,SS11,Ste08a].

Continuous problem.
We consider a general second-order linear system of PDEs where the coefficients . Moreover, we assume that P is coercive on H 1 0 (Ω) D , i.e., the sesquilinear form is elliptic up to some compact perturbation.This is equivalent to strong ellipticity of the matrices A ii ′ in the sense of [McL00,page 119].Here, the standard complex scalar product on C D is denoted by w • z = D j=1 w j z j .Let G : R d \ {0} → C D×D be a corresponding (matrix-valued) fundamental solution in the sense of [McL00,page 198], i.e., a distributional solution of PG = δ, where δ denotes the Dirac delta distribution.For ψ ∈ L ∞ (Γ) D , we define the single-layer operator as (2.13) According to [McL00, page 209 and 219-220] and [HMT09, Corollary 3.38], this operator can be extended for arbitrary σ ∈ (−1/2, 1/2] to a bounded linear operator (2.14) [McL00,Theorem 7.6] states that V is always elliptic up to some compact perturbation.We assume that it is elliptic even without perturbation, i.e.,

Re (Vψ
for all ψ ∈ H −1/2 (Γ) (2.16) Given a right-hand side f ∈ H 1 (Γ) D , we consider the boundary integral equation (2.17)Such equations arise from (and are even equivalent to) the solution of Dirichlet problems of the form Pu = 0 in Ω with u = g on Γ for some g ∈ H 1/2 (Γ) D ; see, e.g., [McL00,] for more details.The Lax-Milgram lemma provides existence and uniqueness of the solution φ ∈ H −1/2 (Γ) D of the equivalent variational formulation of (2.17) (2.18) In particular, we see that V : D is an isomorphism.In the Galerkin boundary element method, the test space H −1/2 (Γ) D is replaced by some discrete subspace Again, the Lax-Milgram lemma guarantees existence and uniqueness of the solution Φ • ∈ X • of the discrete variational formulation (2.19) Moreover, Φ • can in fact be computed by solving a linear system of equations.Note that (2.14) implies that

Axioms of adaptivity (revisited)
The aim of this section is to formulate an adaptive algorithm (Algorithm 3.3) for conforming BEM discretizations of our model problem (2.17), where adaptivity is driven by the residual a posteriori error estimator (see (3.11) below).We identify the crucial properties of the underlying meshes, the mesh-refinement, as well as the boundary element spaces which ensure that the residual error estimator fits into the general framework of [CFPP14] and which hence guarantee optimal convergence behavior of the adaptive algorithm.We mention that we have already identified similar (but not identical) properties for the finite element method in [GHP17, Section 3].The main result of this work is Theorem 3.4 which is proved in Section 4.
3.1.Meshes.Throughout, T • is a mesh of the boundary Γ = ∂Ω of the bounded Lipschitz domain Ω ⊂ R d in the following sense: (i) T • is a finite set of compact Lipschitz domains on Γ, i.e., each element T has the form T = γ T ( T ), where T is a compact 2 Lipschitz domain in R d−1 and γ T : We suppose that there is a countably infinite set T of admissible meshes.In order to ease notation, we introduce for T • ∈ T the corresponding mesh-width function For ω ⊆ Γ, we define the patches of order q ∈ N 0 inductively by The corresponding set of elements is If ω = {z} for some z ∈ Γ, we simply write π q • (z) := π q • ({z}) and Π q • (z) := Π q • ({z}).For S ⊆ T • , we define π q • (S) := π q • ( S) and Π q • (S) := Π q • ( S).To abbreviate notation, the index q = 1 is omitted, e.g., π • (ω) := π 1 • (ω) and Π • (ω) := Π 1 • (ω).We assume the existence of constants C patch , C locuni , C shape , C cent , C semi > 0 such that the following assumptions are satisfied for all T • ∈ T: (M1) Bounded element patch: The number of elements in a patch is uniformly bounded, i.e., (M5) Local seminorm estimate: For all v ∈ H 1 (Γ), it holds that The following proposition shows that (M5) is actually always satisfied.However, in general the multiplicative constant depends on the shape of the point patches.The proof is inspired by [DNPV12, Proposition 2.2], where an analogous assertion for norms instead of seminorms is found.For σ = 1/2 and d = 2, we have already shown the assertion in the recent own work [FGHP16, Lemma 4.5].For polyhedral domains Ω with triangular meshes, it is proved in [FFME + 14, Proposition 3.3] via interpolation techniques.A detailed proof for our setting is found in [Gan17, Proposition 5.2.2],where we essentially follow the proof of [FGHP16, Lemma 4.5].
We suppose that there exist C son ≥ 2 and 0 < ρ son < 1 such that all meshes T • ∈ T satisfy for arbitrary marked elements M • ⊆ T • with corresponding refinement T • := refine(T • , M • ), the following elementary properties (R1)-(R3): (R1) Son estimate: One step of refinement leads to a bounded increase of elements, i.e., #T • ≤ C son #T • , (R2) Father is union of sons: Each element is the union of its successors, i.e., (R3) Reduction of sons: Successors are uniformly smaller than their father, i.e., By induction and the definition of refine(T • ), one easily sees that (R2)-(R3) remain valid if T • is an arbitrary mesh in refine(T • ).In particular, (R2)-(R3) imply that each refined element T ∈ T • \ T • is split into at least two sons, wherefore Besides (R1)-(R3), we suppose the following less trivial requirements (R4)-(R5) with generic constants C clos , C over > 0: (R4) Closure estimate: Let (T ℓ ) ℓ∈N 0 be a sequence in T such that T ℓ+1 = refine(T ℓ , M ℓ ) with some M ℓ ⊆ T ℓ for all ℓ ∈ N 0 .Then, it holds that (R5) Overlay property: For all T • , T ⋆ ∈ T, there exists a common refinement T • ∈ refine(T • ) ∩ refine(T ⋆ ) which satisfies the overlay estimate 3.3.Boundary element space.With each T • ∈ T, we associate a finite dimensional space of vector valued functions We note the Galerkin orthogonality as well as the resulting Céa type quasi-optimality We assume the existence of constants C inv > 0, q loc , q proj , q supp ∈ N 0 , and 0 < ρ unity < 1 such that the following properties (S1)-(S4) hold for all T • ∈ T: Componentwise local approximation of unity: For all T ∈ T • and all j ∈ {1, . . ., D}, there exists some Ψ •,T,j ∈ X • with such that only the j-th component does not vanish, i.e., (Ψ •,T,j ) j ′ = 0 for j ′ = j, and Besides (S1)-(S4), we suppose that there exist constants C sz > 0 as well as q sz ∈ N 0 such that for all T • ∈ T and S ⊆ T • , there exists a linear Scott-Zhang-type operator 3.4.Error estimator.Let T • ∈ T. Due to the regularity assumption f ∈ H 1 (Γ) D , the mapping property (2.14), and This allows to employ the weighted-residual a posteriori error estimator where the local refinement indicators read for all T ∈ T • . (3.11b) The latter estimator goes back to the works [CS96,Car97], where reliability (3.16) is proved for standard 2D BEM with piecewise polynomials on polygonal geometries, while the corresponding result for 3D BEM is found in [CMS01].
3.5.Adaptive algorithm.We consider the following concrete realization of the abstract algorithm from [CFPP14, Algorithm 2.2].

Optimal convergence. Define
and for all s > 0 We say that the solution φ ∈ H −1/2 (Γ) D lies in the approximation class s with respect to the estimator if By definition, φ A est s < ∞ implies that the error estimator η • on the optimal meshes T • decays at least with rate O (#T • ) −s .The following main theorem states that each possible rate s > 0 is in fact realized by Algorithm 3.3.The proof is given in Section 4. It essentially follows by verifying the axioms of adaptivity from [CFPP14].Such an optimality result was first proved in [FKMP13] for the Laplace operator P = −∆ on a polyhedral domain Ω.As ansatz space, they considered piecewise constants on shape-regular triangulations.[FFK + 14] in combination with [AFF + 17] extends the assertion to piecewise polynomials on shaperegular curvilinear triangulations of some piecewise smooth boundary Γ.Independently, [Gan13] proved the same result for globally smooth Γ and general self-adjoint and elliptic boundary integral operators.
Remark 3.6.Let Γ 0 Γ be an open subset of Γ = ∂Ω and let E 0 : L 2 (Γ 0 ) D → L 2 (Γ) D denote the operator that extends a function defined on Γ 0 to a function on Γ by zero.We define the space of restrictions Then, one can consider the integral equation where In the literature, such problems are known as screen problems; see, e.g., [SS11, Section 3.5.3].Theorem 3.4 holds analogously for the screen problem (3.19).Indeed, the works [FKMP13, FFK + 14, AFF + 17, Gan13] cover this case as well.However, to ease the presentation, we focus on closed boundaries Γ 0 = Γ = ∂Ω.
Remark 3.7.(a) Let us additionally assume that X • contains all componentwise constant functions, i.e., x ∈ X • for all x ∈ C D . (3.20) Then, under the assumption that To see this, recall that H 1/2 (Γ) D is continuously and densely embedded in L 2 (Γ) D which is itself continuously and densely embedded in We abbreviate the projection operator J ℓ := J ℓ,T ℓ for all ℓ ∈ N 0 .For all T ∈ T ℓ , the projection property (S5) in combination with our additional assumption (3.20), the triangle inequality, and the local L 2 -stability (S6) show that .
(b) The latter observation allows to follow the ideas of [BHP17] and to show that the adaptive algorithm yields convergence provided that the sesquilinear form (V • ; •) is only elliptic up to some compact perturbation and that the continuous problem is well-posed.This includes, e.g., adaptive BEM for the Helmholtz equation; see [Ste08a, Section 6.9].For details, the reader is referred to [BHP17, BBHP19].
Up to the fact that we allow γ T to be bi-Lipschitz instead of C Let T 0 be a κ 0 -shape regular triangulation.For d = 2, we define refine(•) as in [AFF + 13] via 1D-bisection in the parameter domain.For d = 3, we define refine(•) as in [Ste08b] via newest vertex bisection in the parameter domain.In particular, all corresponding refinements T • ∈ T = refine(T 0 ) are again κ-shape regular triangulations with some fixed κ depending on κ 0 .We also note that the number of different π • (z) in (iv) is uniformly bounded, i.e., there exist only finitely many reference node patches.Finally, let p ∈ N 0 be a fixed polynomial order.For each T • , we associate the space of (transformed) piecewise polynomials For this concrete setting, we already pointed out that [FFK + 14] in combination with [AFF + 17] proved linear convergence (3.17) at optimal rate (3.18) if P = −∆ is the Laplace operator.The following theorem generalizes this result to arbitrary P as in Section 2.3.The inverse inequality (S1) for piecewise polynomials on the boundary is proved, e.g., in [AFF + 17, Lemma A.1]. Nestedness (S2) is trivially satisfied.Also (S3) is trivially satisfied with q loc , q proj = 0. Clearly, (S4) holds with (Ψ •,T,j ) j ′ := 0 for j ′ = j and (Ψ •,T,j ) j := χ T , where χ T denotes the indicator function on T .Finally, for S ⊆ T • ∈ T, we define with the elementwise L 2 (T )-orthogonal projection P This definition immediately yields (S5)-(S6) with q sz = 0.
Remark 3.9.We mention that Theorem 3.8 is also valid if d = 2 and X • is chosen as set of (transformed) splines which are piecewise polynomials with certain differentiability conditions at the break points.The required properties are (implicitly) verified in [FGHP16].As in ), where we have verified the abstract FEM framework of [GHP17] for IGAFEM with hierarchical splines [VGJS11] and the mesh-refinement from [GHP17] (resp.T-splines with the mesh-refinement from [MP15]), the verification of the present abstract BEM framework for 3D IGABEM will be addressed in the future work [GP20].
4. Proof of Theorem 3.4 In the following subsections, we prove Theorem 3.4.Reliability (3.16) is treated explicitly in Section 4.2.It follows immediately from an auxiliary result on the localization of the Sobolev-Slobodeckij norm which is investigated in Section 4.1.To prove Theorem 3.4 (ii)-(iii), we verify the following abstract properties (E1)-(E4) for the error estimator.Together with (R1), the closure estimate (R4), and the overlay property (R5), these already imply linear convergence of the estimator at optimal algebraic rate; see [CFPP14].
Proposition 4.1.Let 0 < σ < 1 and T • ∈ T.Then, (M1) implies the existence of a constant C ′ split > 0 such that for any v ∈ H σ (Γ) D , there holds that The constant C ′ split depends only on the constant from (M1).Proof.With the abbreviation (M1) shows that This concludes the proof.
With this result, one can immediately construct a reliable and efficient error estimator, namely the so-called Faermann estimator ; see Remark 4.10.For d = 2, the result of the proposition goes back to [Fae00], where X • is chosen as space of splines transformed via the arclength parametrization γ : [a, b] → Γ onto the one-dimensional boundary.In the recent own works [FGP15], we generalized the assertion to rational splines, where we could also drop the restriction that γ is the arclength parametrization.For d = 3, [Fae02] proved the result for discrete spaces which contain certain (transformed) polynomials of degree p ∈ {0, 1, 5, 6} on a curvilinear triangulation of Γ.Our proof of Proposition 4.2 is inspired by [Fae02].
The key ingredient is the assumption (S4) which is exploited in Lemma 4.7.Before proving Proposition 4.2, we provide an easy corollary which is the key ingredient for the proof of reliability (3.16).
To prove Proposition 4.2, we start with the following basic estimate, which is proved in [Hac95, Lemma 8.2.4] or in [Gan17, Lemma 5.3.1].
Lemma 4.4.For all λ > 0, there is a constant C(λ) > 0 such that for all x ∈ R d and all ε > 0, there holds that (4.5) The constant C(λ) depends only on the parameter λ, the dimension d, and Γ.
The following lemma is the first step towards the localization of the norm v H σ (Γ) for certain functions v ∈ H σ (Γ) D .In [Fae02, Lemma 3.1], this result is stated for triangular meshes.The elementary proof extends to our situation; see also [Gan17, Lemma 5.3.2]for details.
Lemma 4.5.Let 0 < σ < 1 and T • ∈ T.Then, (M4) implies the existence of a constant C > 0 such that for all v ∈ H σ (Γ) D , it holds that The constant C depends only on the dimension d, σ, Γ, and the constant from (M4).
It remains to control the second summand in (4.6).To this end, we need the following elementary Poincaré type inequality of [Fae00, Lemma 2.5].
Proof.We prove (4.8) for each component v j of v, where j ∈ {1, . . ., D}.Then, squaring and summing up all components, we conclude the proof.(S4) and Lemma 4.6 show that (4.9) Now, we apply the orthogonality and (S4) to get for the second summand that Inserting this in (4.9) gives that T .Inserting this in (4.10) and using again (S4), we derive that )) .Altogether, this concludes the proof.
The following lemma allows us to further estimate the term |v| H σ (π qsupp • (T )) of (4.8).Lemma 4.8.Let q ∈ N 0 and T • ∈ T.Then, (M1)-(M4) imply the existence of a constant C(q) > 0 such that for all v ∈ H σ (Γ) D and all T ∈ T • there holds that The constant depends only on the dimension d, σ, q, and the constants from (M1)-(M4).
Proof.Without loss of generality, we may assume that D = 1.We prove the assertion in two steps.

This implies that
Together with the induction hypothesis (4.13), this concludes the induction step.
Step 2: We come to the assertion itself.By definition, we have that Let T ′ , T ′′ ∈ Π q • (T ).First, we suppose that T ′ = T ′′ = ∅.Then, there exists a chain as in Step 1 with T ′ = T 0 and T ′′ = T m .Step 1 proves that If T ′ = T ′′ , the same estimate holds true.Since the number of T ′ , T ′′ ∈ Π q • (T ) is uniformly bounded by a constant, which depends only on the constant of (M1) and q, this estimate concludes the proof.
Proof.We apply Lemma 4.7 and Lemma 4.8 to see that For T ′ , T ′′ ∈ T • with T ′ ∩ T ′′ = ∅, we fix some point z(T ′ , T ′′ ) ∈ T ′ ∩ T ′′ .With (M5), we continue our estimate (T ), and that the last term of the latter estimate can be bounded from above (up to a multiplicative constant) by h . This concludes the proof.With all the preparations, we can finally prove the main result of this section.
Proof of Proposition 4.2.Together with (M3), Lemma 4.5 proves that It remains to estimate the second sum.With Lemma 4.7 and Lemma 4.8, we see that ). Plugging this into (4.16)shows that (4.17) Remark 4.10.Proposition 4.1 and Proposition 4.2 show that This is even true for arbitrary f ∈ H 1/2 (Γ) D without the additional restriction f ∈ H 1 (Γ) D .In particular, provides a local error indicator.The corresponding error estimator ̥ • is often referred to as Faermann estimator.In BEM, it is the only known estimator which is reliable and efficient (without further assumptions as, e.g., the saturation assumption [FLP08, Section 1]).Obviously, one could replace the residual estimator η ℓ in Algorithm 3.3 by ̥ ℓ .However, due to the lack of an h-weighting factor, it is unclear whether the reduction property (E2) of Section 4.2 is satisfied.[FFME + 14, Theorem 7] proves at least plain convergence of ̥ ℓ even for f ∈ H 1/2 (Γ) D if one uses piecewise constants on affine triangulations of Γ as ansatz space.
The proof immediately extends to our current situation, where the assumptions (M1)-(M5), (R2)-(R3), and (S1)-(S2) are employed.The key ingredient is the construction of an equivalent mesh-size function h • ∈ L ∞ (Γ) which is contractive on each element which touches a refined element, i.e., there exists a uniform constant 0 < ρ ctr < 1 such that The existence of such a mesh-size function is proved in [CFPP14, Section 8.7] for shaperegular triangular meshes.The proof works verbatim for the present setting.

Convergence of Φ
and hence admits a unique Galerkin solution Φ ∞ ∈ X ∞ .Note that Φ ℓ is also a Galerkin approximation of Φ ∞ .Hence, the Céa lemma (3.10) with φ replaced by Φ ∞ proves that Φ ∞ − Φ ℓ H −1/2 (Γ) → 0 as ℓ → ∞.In particular, we obtain that lim ℓ→∞ Φ ℓ+1 − Φ ℓ H −1/2 (Γ) = 0. 4.4.An inverse inequality for V.In Proposition 4.13, we establish an inverse inequality for the single-layer operator V. Throughout this section, the ellipticity of V is not exploited (and we can drop this assumption here).For the Laplace operator P = −∆, such an estimate was already proved in [FKMP13, Theorem 3.1] for shape-regular triangulations of a polyhedral boundary Γ.Independently, [Gan13] derived a similar result for globally smooth Γ and arbitrary self-adjoint and elliptic boundary integral operators.In [AFF + 17, Theorem 3.1], [FKMP13, Theorem 3.1] is generalized to piecewise polynomial ansatz functions on shape-regular curvilinear triangulations.In particular, our Proposition 4.13 does not only extend these results to arbitrary general meshes as in Section 3.1, but is also completely novel for, e.g., linear elasticity.The proof follows the lines of [AFF + 17, Section 4].We start with the following lemma, which was proved in [CP06, Theorem 4.1] on shape-regular triangulations.With Lemma 4.6, the proof immediately extends to our situation; see also [Gan17,Lemma 5.3.11].
Lemma 4.11.For T • ∈ T, let P 0 (T • ) D ⊂ L 2 (Γ) D be the set of all functions whose D components are T • -piecewise constant functions on Γ.Let P • : L 2 (Γ) D → P 0 (T • ) D be the corresponding L 2 -projection.Then, (M1) and (M3) imply for arbitrary 0 < σ < 1 the existence of a constant C > 0 such that The constant C depends only on the dimension D, the boundary Γ, σ, and the constants from (M3).
In contrast to [AFF + 17], we cannot use the Caccioppoli type inequality from [Mor08, Lemma 5.7.1] which is only shown for the Poisson problem there.Therefore, we prove the following generalization.For an open set O ⊂ R d and an arbitrary u ∈ H 2 (O), we abbreviate Lemma 4.12.Let r > 0, x ∈ R d , and u ∈ H 1 (B 2r (x)) D be a weak solution of Pu = 0.Then, u| Br(x) ∈ C ∞ (B r (x)) D and there exists a constant C > 0 such that The constant C depends only on the dimensions d, D, and the coefficients of the partial differential operator P.
The application of [McL00, Theorem 4.16] yields the existence of a constant C 1 > 0, which depends only on d, D, and the coefficients of the matrices A ii ′ , such that Standard scaling arguments prove that Plugging this into (4.24),we obtain that We choose λ as the integral mean λ := B 3r/2 (x) u(y)dy/|B 3r/2 (x)|.The Cauchy-Schwarz inequality implies that Using this and the Poincaré inequality in (4.25), we see that Together with the fact that B 3r/2 (x) ⊂ B 2r (x), this concludes the proof.
For the proof of the next proposition, we need the linear and continuous single-layer potential from [McL00, Theorem 6.11] where U is an arbitrary bounded domain with Γ ⊂ U.The single-layer operator V is just the trace of V, i.e., Then, there exists a constant C inv,V > 0 such that for all ψ ∈ L 2 (Γ) D , it holds that The constant C inv,V depends only on (M1)-(M5), Γ, the coefficients of P, and the admissibility constant α.The particular choice w Proof.The proof works essentially as in [AFF + 17, Section 4].Therefore, we mainly emphasize the differences and refer to [Gan17, Proposition 5.3.15]for further details.By (M4) and with the abbreviation δ 1 (T ) := diam(T )/(2C cent ) and . This provides us with an open covering of Γ ⊂ T ∈T• U T .We show that this is even locally finite in the sense that there exists a constant C > 0 with # T ∈ T Clearly, we may assume that x ∈ T ∈T• U T .Choose T 0 ∈ T • with x ∈ U T 0 such that δ 1 (T 0 ) is minimal, and let x 0 ∈ T 0 with |x − x 0 | < δ 1 (T 0 ).If T ∈ T • with x ∈ U T , the triangle inequality yields that dist({x 0 }, T ) < 2δ 1 (T ).By choice of δ 1 (T ), (M4) hence yields that We fix (independently of T • ) a bounded domain U ⊂ R d with U T ⊂ U for all T ∈ T • .We define for T ∈ T • the near-field and the far-field of u V := Vψ by u near V,T := V(ψχ Γ∩U T ) and u far V,T := V(ψχ Γ\U T ).(4.33) In the following five steps, the near-field and the far-field are estimated separately.The first two steps deal with the near-field, whereas the last three steps deal with the far-field.
Step 1: As in [AFF + 17, Lemma 4.1], one shows that that for all Step 2: With Step 1, one shows as in [AFF + 17, Proposition 4.2] that u near V,T ∈ H 1 (U) and In the proof, one applies the stability of V : L 2 (Γ) D → H 1 (Γ) D (see (2.14)) and V : Step 3: We consider the far-field.We set Ω ext := R d \ Ω.According to [McL00, Theorem 6.11], for all T ∈ T • , the potential u far V,T is a solution of the transmission problem where [•] Γ and [D ν (•)] Γ denote the jump of the traces and the conormal derivatives respectively (see [McL00,page 117] for a precise definition) across the boundary Γ.Twofold integration by parts that uses these jump conditions shows that Pu far V,T = 0 weakly on U T .Since B δ 1 (T ) (x) ⊆ U T for all x ∈ T , Lemma 4.12 shows that u far Note that [AFF + 17] proves (4.36) even without the term u far V,T L 2 (B δ 1 (T ) (x)) .Indeed, since the kernel of the Laplace operator contains all constants, [AFF + 17] employs a Poincaré inequality to bound u far Step 4: With inequality (4.36) at hand, one can prove the following local far-field bound for the single-layer potential V The proof works as in [AFF + 17, Lemma 4.4], and relies on a standard trace inequality on Ω, the Caccioppoli inequality (4.36) as well as the Besicovitch covering theorem.Note that in [AFF + 17], the estimates (4.36) and thus (4.37) even hold without the L 2 -norm of u far V,T , since the Laplace problem is considered.
Step 5: Finally, (4.32), (4.33), (4.35), (4.37), and the stability of V : H −1/2 (Γ) D → H 1 (U) D (see (4.26)) easily lead to the far-field bound for V [AFF + 17] does not only treat the single-layer operator V : H −1/2 (Γ) D → H 1/2 (Γ) D , but also derives similar inverse estimates as in (4.30) for the double-layer operator, the adjoint double-layer operator, and the hyper-singular operator.With similar techniques as in Proposition 4.13, we will also generalize this result in Appendix B. However, we will indeed only need the inverse estimate (4.31) for the single-layer operator in the remainder of the paper.
4.8.Discrete reliability (E4).The proof of (E4) is inspired by [FKMP13, Proposition 5.3] which considers piecewise constants on shape-regular triangulations as ansatz space.Under the assumptions (M1)-(M5), (3.6), and (S1)-(S6), we show that there exist C drel , C ref ≥ 1 such that for all T • ∈ T and all T • ∈ refine(T • ), the subset satisfies that The last two properties are obvious with by validity of (M1) and (3.6).The first estimate is proved in three steps: Step 1: For S 1 := T • ∩ T • , let J •,S 1 be the corresponding projection operator from (S5)-(S6).Ellipticity (2.15), nestedness (S2) of the ansatz spaces, and the definition (3.8) of the Galerkin approximations yield that Hence, the local projection property (S5) of J •,S 1 is applicable and proves that J • , where the hidden constant depends only on d and (M1)-(M4).This leads to (4.47) We bound the two terms I : The Cauchy-Schwarz inequality shows that . This together with the fact that h Step 2: Next, we deal with the first summand of (4.50).With supp(χ) .
(4.51)By Galerkin orthogonality (3.9), we see that V(φ − Φ • ) is L 2 -orthogonal to all functions of X • which includes in particular the functions Ψ •,T,j from (S4).Hence, we can apply Proposition 4.9.Together with (M1)-(M3) and recalling (4.45), (4.51) proves that Step 3: It remains to consider the second summand of (4.50).Lemma 4.5 in conjunction with shape-regularity (M3) implies that We have already dealt with the second summand in Step 2 (see (4.51)).For the first one, we fix again some z(T, T With the product rule and (A.2), we continue our estimate . Note that we have already dealt with the first summand in Step 2 (see (4.51)).Finally, supp(χ With this, we conclude the proof of discrete reliability (E4).The constant C drel depends only on C ̺ , d, D, Γ, and the constants from (M1)-(M5) and (S1)-(S6).
Step 3: Finally, we prove (A.2).We recall that δ • > 0 on O; see Step 1.With the identity matrix I ∈ R d×d and the matrix (x − y)(∇δ • (x)) ⊤ ∈ R d×d , elementary calculations prove for all x ∈ O ⊃ Γ and all y ∈ R d that Considering the norm, we see that Together with supp(K s ) = B s (0), this shows for all x ∈ Γ that Thus, (A.5)-(A.6)and (M3) prove that |∇ χ S (x)| h • (x) −1 for almost all x ∈ Γ.Moreover, for smooth functions, the surface gradient ∇ Γ is the orthogonal projection of the gradient ∇ onto the tangent plane; see, e.g., [ME14, Lemma 2.22]).With the outer normal vector ν, this implies that ∇ Γ χ S = ∇ χ S − (∇ χ S • ν)ν almost everywhere on Γ, and concludes the proof with the previous estimate.
Step 5: To deal with the third summand in (B.10), we first rewrite it as follows To deal with the conormal derivative, we apply again (B.1) together with the fact that V is a potential, i.e., P V = 0 weakly.This leads to The first summand can be bounded as in (B.13).For the second summand, we use the stability (4.26) in combination with • H −1/2 (Γ) • L 2 (Γ) and (B.11).
Finally, it only remains to bound the trace as well as the conormal derivative of the second summand in (B.12).Choosing σ = −1 in the additional regularity (B.9), one sees as in the proof of [McL00,page 203] (which proves stability of V : H −1/2 (Γ) D → H 1 (Ω) D ) that Overall, we have estimated the trace as well as the conormal derivative of all terms in (B.10).Since K − K 0 = ( K − K 0 )(•)| Γ and W − W 0 = −D ν ( K − K 0 ), this verifies the stability (B.7) and thus concludes the proof.