Adaptive boundary element methods with convergence rates
AAdaptive boundary element methods withconvergence rates
Tsogtgerel Gantumur
McGill University
Abstract
This paper presents adaptive boundary element methods for positive, negative, as well as zero orderoperator equations, together with proofs that they converge at certain rates. The convergencerates are quasi-optimal in a certain sense under mild assumptions that are analogous to what istypically assumed in the theory of adaptive finite element methods. In particular, no saturation-type assumption is used. The main ingredients of the proof that constitute new findings are someresults on a posteriori error estimates for boundary element methods, and an inverse-type inequalityinvolving boundary integral operators on locally refined finite element spaces.
Let Γ be a (closed or open) polyhedral surface in R . We consider equations of the form Au = f, (1)where f ∈ H − t (Γ), and A : H t (Γ) → H − t (Γ) is an invertible linear operator. Strictlyspeaking, the function spaces should be slightly modified if Γ is an open surface or t >
0, butfor the sake of this introduction we will gloss over this point. The operators of interest arethe boundary integral operators that arise from reformulations of boundary value problemsas integral equations on the domain boundary, cf. McLean (2000). Then the problem (1)corresponds to a boundary integral equation , and a very popular class of methods for itsnumerical solution is boundary element methods (BEM), which can crudely be describedas finite element methods (FEM) applied to boundary integral equations, cf. Sauter andSchwab (2011). As with finite elements, there is the adaptive version of BEM, whose mainfeature is to automatically distribute mesh points hopefully in an optimal way so as toobtain an accurate numerical solution. Even though those methods perform very well inpractice, their mathematical theory is not in a very satisfactory state, especially if onecompares it with the corresponding theory of adaptive FEM. In the latter context, thesequence of papers D¨orfler (1996), Morin, Nochetto, and Siebert (2002), Binev, Dahmen,and DeVore (2004), Stevenson (2007), and Cascon, Kreuzer, Nochetto, and Siebert (2008)1 a r X i v : . [ m a t h . NA ] D ec Introduction 2 has laid a fairly satisfactory foundation to mathematical understanding of adaptive FEMfor linear elliptic boundary value problems. Specifically, it was established that standardadaptive FEM generate a sequence of solutions whose error decreases geometrically, andthat the number of triangles in the mesh grows with an optimal rate. Moreover, rigoroustreatments of numerical integration and linear algebra solvers are within reach.In parallel to the above, a very general theory of adaptive wavelet methods has beendeveloped, cf. Cohen, Dahmen, and DeVore (2001, 2002); Gantumur, Harbrecht, andStevenson (2007). Under this framework, one can analyze convergence rates and complexityof fully discrete adaptive wavelet methods for boundary integral equations, cf. Stevenson(2004); Gantumur and Stevenson (2006); Dahmen, Harbrecht, and Schneider (2007)However, there has been a significant gap in the current mathematical understand-ing of adaptive BEM proper. The first steps toward closing the gap have been taken inCarstensen and Praetorius (2012), where convergence is guaranteed for an adaptive BEMwith a feedback control that occasionally adds uniform refinements, in Ferraz-Leite, Ort-ner, and Praetorius (2010) where geometric error reduction is proven under a saturationassumption , and in Aurada, Ferraz-Leite, and Praetorius (2012b), where convergence is es-tablished under a weak saturation-type assumption. To give an idea of what a saturationassumption is, one form of it requires that if u k is the numerical solution of (1) at the cur-rent stage of the algorithm execution, and if ˆ u k is the would-be numerical solution had wereplaced the current mesh by its uniform refinement, then (cid:107) u − u k (cid:107) H t (Γ) ≤ β (cid:107) ˆ u k − u k (cid:107) H t (Γ) ,where β > t > t < a posteriori errorestimators for BEM and an inverse-type inequality involving boundary integral operatorsand locally refined meshes. The issue with the theory of error estimators has been most Introduction 3 obvious; what is usually guaranteed is only one of the two inequalities that are necessaryfor a quantity to resemble the error. There is a very few estimators with both upper andlower bounds proven; one can mention the estimators proposed in Faermann (1998, 2000,2002), see Carstensen and Faermann (2001) for a more thorough discussion. To namesome of the relatively recent works on this subject, we have Carstensen, Maischak, andStephan (2001), Carstensen, Maischak, Praetorius, and Stephan (2004), and Nochetto, vonPetersdorff, and Zhang (2010), where upper bounds for certain residual-type estimators areestablished, and Erath, Ferraz-Leite, Funken, and Praetorius (2009a), and Erath, Funken,Goldenits, and Praetorius (2009b), where upper and lower bounds are proven for a largenumber of non-residual type estimators, with the upper bounds depending on various formsof saturation assumptions. Now, even if one had both upper and lower bounds for an errorestimator, one still needs a so-called local discrete bound before attempting to apply thegeneral techniques from the finite element theory. Such an estimate has been entirely openfor boundary element methods.In this work, we establish all missing bounds for a number of residual-type error estima-tors for positive, negative, as well as zero order boundary integral equations, including theestimators from Carstensen et al. (2001), Carstensen et al. (2004), and Faermann (2000,2002). The recent papers Feischl et al. (2011a,b) also contain similar results regarding theestimator from Carstensen et al. (2001). Some of our bounds involve the so-called oscil-lation terms , that give useful estimates on how far the current mesh is from saturation.Note that analogous terms also arise in the finite element theory. In order to handle theoscillation terms, which turned out to be not straightforward, we prove an inverse-typeinequality involving boundary integral operators and locally refined meshes. Our proofof the inverse-type inequality in general requires the underlying surface Γ to be C , orsmoother, but for open surfaces it allows the boundary of Γ to be Lipschitz. So in general,polyhedral surfaces are ruled out. However, this is very likely an artifact of the proof, sincein Feischl et al. (2011a,b), the inequality is proven for a model negative order operator onpolyhedral surfaces.This paper is organized as follows. In the next section, we fix the general setting ofthe paper, and recall some basic results that will be used throughout the paper. Thenin the following three sections, namely in § §
4, and §
5, we consider adaptive BEMs forzero, positive, and negative order operator equations, respectively. In each of the threecases, we first study certain residual based a posteriori error estimators, then design anadaptive BEM based on those estimators, and finally address the question of convergencerate. There are some general arguments and remarks that can be applied to all of the threecases, and so in order not to be too repetitive, they shall be discussed in § § §
5, we will simply refer to them if needed. The entireanalysis depends on an inverse-type inequality involving boundary integral operators andlocally refined meshes, which then is verified in § Generalities 4
In this section, we will set up the necessary vocabulary and collect some basic results thatwill be used throughout the paper. Let Ω be a compact closed, n -dimensional, patchwisesmooth, globally C ν − , manifold, which will be the habitat of all functions and distribu-tions that ever occur in this paper. In practice, we typically have n = 1 or n = 2, so in thediscussions that follow we often use the language of n = 2, and we shall indicate wheneverwe lose generality by implicitly restricting to the case n = 2. We will assume that eachsmooth (closed) patch of Ω is diffeomorphic to a polygon, so one can think of Ω as thesurface of a bounded polyhedron, with faces and edges now allowed to be smooth surfacesand curves, respectively. For s ∈ [ − ν, ν ], we denote by H s (Ω) the usual Sobolev space of order s on Ω. Let ω ⊆ Ωbe an open subset of Ω with Lipschitz boundary. Then we define the following two kindsof Sobolev spaces H s ( ω ) = { u | ω : u ∈ H s (Ω) } , and ˜ H s ( ω ) = { u ∈ H s (Ω) : supp u ⊆ ω } , (2)for s ∈ [ − ν, ν ]. Obviously ˜ H s (Ω) = H s (Ω), and it is known that ˜ H s ( ω ) ∗ = H − s ( ω ),cf. McLean (2000). Note also that ˜ H s ( ω ) is a closed subspace of H s ( ω ) for s ≥
0. Thedefinitions (2) give rise to the canonical norms on H s ( ω ) and ˜ H s ( ω ) inherited from the normon H s (Ω), and at least for s >
0, these norms are known to be equivalent to certain normsdefined either by interpolation, or in terms of moduli of smoothness, with the equivalenceconstants depending only on the dimension n , the order s , the Lipschitz constant of ω ,and the particulars (i.e., the local coordinate patches and the partition of unity) in thedefinition of H s (Ω). Let us expand on this a bit. Since the spaces with s > s ∈ [0 , | s | ≤
1. With (cid:107) · (cid:107) ω and | · | ,ω denoting the L -norm and the usual H -seminorm on ω , respectively, we define | · | H s ( ω ) to be the interpolatory seminorm between (cid:107) · (cid:107) ω and | · | ,ω , for concreteness by using the K -functional. Then (cid:107) · (cid:107) ˜ H s ( ω ) := | · | H s ( ω ) is anorm on ˜ H s ( ω ), which can be made into a norm (cid:107) · (cid:107) H s ( ω ) on H s ( ω ) by combining it withthe L -norm. It can be shown that these norms are equivalent to the canonical norms on˜ H s ( ω ) and respectively H s ( ω ), with equivalence constants depending only on n , s , and theparticulars of the definition of H s (Ω), cf. McLean (2000). Moreover, we have the followinguseful property that if ω , . . . , ω k are disjoint Lipschitz domains such that (cid:83) i ω i = ω , thenfor | s | ≤ (cid:88) i (cid:107) u (cid:107) H s ( ω i ) ≤ C s (cid:107) u (cid:107) H s ( ω ) , and (cid:107) v (cid:107) H s ( ω ) ≤ C s (cid:88) i (cid:107) v (cid:107) H s ( ω i ) , (3) Generalities 5 for u ∈ H s ( ω ) and v ∈ ˜ H s ( ω ), with the constant C s > s , and thenorms for s < H s ( ω ) for s ∈ (0 , Slobodeckijnorm (cid:107) v (cid:107) s,ω = (cid:107) v (cid:107) ω + | v | s,ω , with | v | s,ω = (cid:90) (cid:90) ω × ω | v ( x ) − v ( y ) | | x − y | s d x d y, (4)where (cid:107) v (cid:107) ω denotes the L -norm on ω . It is immediate that if ω , . . . , ω k are disjoint andLebesgue measurable sets such that (cid:83) i ω i ⊆ ω , then for s > (cid:88) i | v | s,ω i ≤ | v | s,ω , v ∈ H s ( ω ) , (5)which is the counterpart of the first inequality in (3). The seminorms | · | H s ( ω ) and | · | s,ω areequivalent, with the equivalence constants depending only on n , s , the Lipschitz constantof ω , and as usual the particulars of the definition of H s (Ω). This fact can be proven asin McLean (2000) by relating the both seminorms to the canonical norms defined throughthe norm on H s (Ω). A more direct way to relate the two seminorms would be to firstconnect the Slobodeckij seminorm with the Besov-style seminorm defined by moduli ofcontinuity, and then use the equivalence between the moduli of continuity and the K -functional, established in Johnen and Scherer (1977). Note that one has to inspect (andadapt) the proofs in McLean (2000) and Johnen and Scherer (1977) to reveal the relevantinformation on the equivalence constants. It should be emphasized that since we shallbe dealing with an infinite collection of domains, in partiuclar of sizes that can shrink tozero, when using norm equivalences one must be careful about ensuring a control over theequivalence constants. In the current setting, the equivalence constants do not depend onthe size of ω , and the Lipschitz constants are controlled by restricting the class of domainsto shape regular triangles, see below (14). This can also be seen directly from the fact thatour refinement procedures lead to only finitely many equivalence classes of triangles. Inthe proofs, we make an effort to use the interpolatory norms for as long as possible, andso to apply the norm equivalence only when necessary. In the following, we fix Γ ⊆ Ω to be the whole of Ω, or a connected open set whoseboundary consists of curved polygons. This will be the domain on which we consider ourmain operator equation Au = f, (6)where f ∈ H − t (Γ), and A : ˜ H t (Γ) → H − t (Γ) is a linear homeomorphism. We assume thatthe operator A is self-adjoint and satisfies (cid:104) Av, v (cid:105) ≥ α (cid:107) v (cid:107) H t (Γ) , (cid:107) Av (cid:107) H − t (Γ) ≤ β (cid:107) v (cid:107) ˜ H t (Γ) , (7) Generalities 6 for v ∈ ˜ H t (Γ), with some constants α > β >
0, where (cid:104)· , ·(cid:105) is the duality pairingbetween H − t (Γ) and ˜ H t (Γ). We introduce the energy norm ||| · ||| = (cid:104) A · , ·(cid:105) / , and note thatit is equivalent to the ˜ H t (Γ)-norm √ α (cid:107) · (cid:107) ˜ H t (Γ) ≤ ||| · ||| ≤ (cid:112) β (cid:107) · (cid:107) ˜ H t (Γ) . (8)We also have the norm equivalence α (cid:107) · (cid:107) ˜ H t (Γ) ≤ (cid:107) A · (cid:107) H − t (Γ) ≤ β (cid:107) · (cid:107) ˜ H t (Γ) , (9)which is the basis of all residual based error estimation techniques.Suppose that a closed linear subspace S ⊂ ˜ H t (Γ) is given. Then the Galerkin approxi-mation u S ∈ S of u from the space S is characterized by (cid:104) Au S , v (cid:105) = (cid:104) f, v (cid:105) , ∀ v ∈ S. (10)We have the Galerkin orthogonality ||| u − u S ||| + ||| u S − v ||| = ||| u − v ||| , ∀ v ∈ S, (11)which implies that u S is the best approximation of u from S in the energy norm, and thatthe Galerkin approximation is stable: √ α (cid:107) u S (cid:107) ˜ H t (Γ) ≤ ||| u S ||| ≤ ||| u ||| ≤ √ α (cid:107) f (cid:107) H − t (Γ) . (12)In light of (9), the residual r S = f − Au S is equivalent to the error: α (cid:107) u − u S (cid:107) ˜ H t (Γ) ≤ (cid:107) r S (cid:107) H − t (Γ) ≤ β (cid:107) u − u S (cid:107) ˜ H t (Γ) . (13)In the sense that the residual is a computable quantity that gives bounds on the true errorin terms of this equivalence, the first inequality in (13) is an example of a global upperbound , while the second one is that of a global lower bound . Upper and lower bounds inthis context are also called reliability and efficiency , respectively. The central issue in thetheory of residual based error indicators is to somehow localize the quantity (cid:107) r S (cid:107) H − t (Γ) soas to obtain a useful information on which part of Γ needs more attention. We study the Galerkin approximation by piecewise constant or continuous piecewise linearfunctions on adaptively generated triangulations of the manifold Γ. Let us now fix somenotations and terminologies related to this discretization. An open subset of Γ is called a (surface) triangle if its closure is diffeomorphic to a flat triangle, and the latter is said to bethe reference of the former. Assuming that a choice is made of a reference for each surface
Generalities 7 triangle, notions related to flat triangles can be planted onto surface triangles throughtheir references. For instance, in the following, straight lines, midpoint, etc., should beunderstood in terms of the reference triangles. We call a collection P of surface trianglesa partition of Γ if Γ = (cid:83) τ ∈ P τ , and τ ∩ σ = ∅ for any two different τ, σ ∈ P . Forrefining the meshes we mainly use the so called newest vertex bisection algorithm, whichwe describe now for the reader’s convenience. General discussions on this algorithm canbe found e.g., in Binev, Dahmen, and DeVore (2004); Stevenson (2008). We assume thatwith any triangle τ comes its associated newest vertex v ( τ ), so that when it is needed tobe refined, τ is subdivided into two triangles by connecting v ( τ ) with the midpoint of theedge opposite to it. The midpoint used in the bisection is now the newest vertex of theboth new triangles. The 2 new triangles so obtained are called the children of τ , and therefinement of τ is just the collection of its children. The children of a triangle inherit thereference map from their parent. A partition P (cid:48) is called a refinement of P and denoted P (cid:22) P (cid:48) if P (cid:48) can be obtained by replacing zero or more τ ∈ P by its children, or by arecursive application of this procedure. This procedure is extended to higher dimensionsin Stevenson (2008).A partition P is said to be conforming if any vertex v of a triangle in P is a vertexof all τ ∈ P whose closure contains v . Throughout this paper we consider only partitionsthat are refinements of some fixed conforming partition P of Γ. We require that thereferences of the initial partition P be so that for any pair of surface triangles that share acommon edge, the parameterizations from the reference triangles to the common edge areequal up to the composition with an affine map. The motivation for this is that we wantthe refinements on both triangles to agree on the common edge. A choice of refinementprocedure immediately leads to the set [ P ] of all partitions that are refinements of P . Weassume that the family [ P ] is shape regular , meaning that σ s = sup (cid:26) h nτ vol( τ ) : τ ∈ P, P ∈ [ P ] (cid:27) < ∞ , (14)where h τ = diam( τ ). Both the newest vertex bisection and the red refinement proceduresproduce shape regular partitions. The set [ P ] is too large in the sense that often weare interested in a certain subset of it that has a good analytic property, e.g., we wantto single out the conforming partitions from [ P ]. Exactly what subset we want dependson the particular setting, and at this level of generality we simply assume that there is asubset adm( P ) ⊂ [ P ] called the family of admissible partitions , which is graded (or locallyquasi-uniform , or have the K-mesh property ), i.e., σ g = sup (cid:26) h σ h τ : σ, τ ∈ P, σ ∩ τ (cid:54) = ∅ , P ∈ adm( P ) (cid:27) < ∞ . (15)For example, if n ≥ P produced by the newestvertex bisection are locally quasi-uniform. If n = 1, we define the admissible partitions Generalities 8 to be the ones for which the quantity under the supremum in (15) is bounded by a fixednumber. Finally, note that the shape regularity and local quasi-uniformity together imply local finiteness , meaning that the number of triangles meeting at any given point is boundedby a constant that depends only on σ s , σ g , and n .The admissible partitions are the only ones that are “visible” to the analytic com-ponents of the algorithms. Hence from both analytic and algorithmic perspectives, it isconvenient to separate the “analytic” components that only see admissible partitions, fromthe “combinatoric” components that make possible the illusion that there are only admis-sible partitions. These “combinatoric” issues are common to both the FEM and BEM,and mostly settled. We take them into account by assuming the existence of a couple ofoperations on admissible partitions. The first operation is that of refinement, which inpractice is implemented by a usual naive refinement possibly producing a non-admissiblepartition, followed by a so-called completion procedure. Given a partition P ∈ adm( P )and a set R ⊂ P of its triangles, the refinement procedure produces P (cid:48) ∈ adm( P ), suchthat P \ P (cid:48) ⊇ R , i.e., the triangles in R are refined at least once. Let us denote it by P (cid:48) = refine( P, R ). We assume the following on its efficiency: If { P k } ⊂ adm( P ) and { R k } are sequences such that P k +1 = refine( P k , R k ) and R k ⊂ P k for k = 0 , , . . . , then P k − P ≤ C c k − (cid:88) m =0 R m , k = 1 , , . . . , (16)where C c > overlay of partitions: We assume that there is anoperation ⊕ : adm( P ) × adm( P ) → adm( P ) satisfying P ⊕ Q (cid:23) P, P ⊕ Q (cid:23) Q, and P ⊕ Q ) ≤ P + Q − P , (17)for P, Q ∈ adm( P ). In the conforming world, P ⊕ Q is taken to be the smallest and commonconforming refinement of P and Q , for which (17) is demonstrated in Cascon et al. (2008).For a 1D refinement procedure, a justification is given in Aurada et al. (2012a). Given a partition P ∈ [ P ], we define the piecewise polynomial space S dP by S dP = (cid:8) u ∈ L (Γ) : u ∈ C (Γ) if d > , and u | τ ∈ P d ∀ τ ∈ P (cid:9) , (18)where P d denotes the set of polynomials of degree less than or equal to d . Note thatfor curved triangles, polynomials are defined through the reference triangles. We have S dP ⊂ H s (Γ) for s < if d = 0, and s < if d >
0, with | s | ≤ ν in both cases. Of Generalities 9 interest to us are only the spaces S P of piecewise constants and S P of continuous piecewiselinears. We will also employ a slight variation of S P , that is the space ˜ S P of piecewiseaffine functions that vanish on the boundary of Γ.Now we collect some estimates relating different Sobolev norms for finite element spacesand their complements. We will indicate if the constants involved in the estimates dependon parameters (such as σ s ) other than d and n . First of all, we recall Faermann’s estimate (cid:107) v (cid:107) s, Γ ≤ C F (cid:88) z ∈ N P | v | s,ω ( z ) , v ∈ H s (Γ) , v ⊥ L S P , ( s ∈ [0 , , (19)for all admissible partitions P ∈ adm( P ), with C F = C F ( σ s , σ g ), cf. Faermann (2000,2002). Here N P is the set of vertices in P , and ω ( z ) = int (cid:83) { τ ∈ P : z ∈ τ } τ is called the star associated to the vertex z , with “int” denoting the interior. The same estimate forinterpolatory norms has been established in Carstensen, Maischak, and Stephan (2001),by a very flexible technique. We will be using their technique on several occasions in § P ∈ [ P ], let h P ∈ S P be such that h P ( x ) = h τ for x ∈ τ ∈ P .We introduce the space H r (Γ , P ) for r > v ∈ L (Ω) with v | τ ∈ H r ( τ ) for every triangle τ ∈ P . Now for v ∈ H s (Γ , P ) with s ∈ [0 , v P ∈ S P be the L -orthogonal projection of v onto S P . Then we have the direct estimate (cid:107) v − v P (cid:107) τ ≤ C J h sτ | v | s,τ , τ ∈ P, (20)where the constant C J = C J ( σ s ). An immediate consequence is that (cid:107) h rP ( v − v P ) (cid:107) γ ≤ C J (cid:88) τ ∈ Q h r + s ) τ | v | s,τ , (21)for Q ⊆ P , γ = int (cid:83) τ ∈ Q τ , and r ∈ R . By using a duality argument, and the bounds (21)and (5), one can show also that (cid:107) v − v P (cid:107) ˜ H − s ( γ ) ≤ C J (cid:107) h sP v (cid:107) γ , v ∈ L (Γ) . (22)For the continuous piecewise linears, the L -projection is nonlocal, and so for conve-nience we will employ a quasi-interpolation operator Q P : L (Γ) → ˜ S P that satisfies (cid:107) v − Q P v (cid:107) r,γ ≤ C J (cid:18) max τ ∈ Q h τ (cid:19) s − r | v | s,ω ( γ ) , v ∈ ˜ H s (Γ) , (23)for 0 ≤ r ≤ s ≤ P ∈ adm( P ), where ω ( γ ) = int (cid:83) { σ ∈ P : σ ∩ γ (cid:54) = ∅ } σ , and if n ≥ P ) is understood to be the conforming partitions created by newest vertex bisectionsfrom P . Recall also that ˜ S P is the subspace of S P with the homogeneous boundarycondition. By a quasi-interpolation we mean that ( Q P v ) | τ = ( Q P v | ω ( τ ) ) | τ . Examples ofsuch operators are constructed, e.g., in Cl´ement (1975); Scott and Zhang (1990); Oswald Operators of order zero 10 (1994); Bernardi and Girault (1998). Accordingly, in this setting, the estimate (21) isreplaced by (cid:107) h rP ( v − Q P v ) (cid:107) γ ≤ C J (cid:88) τ ∈ Q h r + s ) τ | v | s,ω ( τ ) . (24)We stated the estimates (20) and (23) in terms of the Slobodeckij norms, but the usualway to derive these estimates is by interpolation and norm equivalences, so in particularthe same estimates hold with interpolatory norms.Let us also recall the inverse estimates (cid:107) v (cid:107) H s ( γ ) ≤ C B (cid:107) h − sP v (cid:107) γ , v ∈ S dP , (25)for s ∈ [0 , ) ∩ [0 , ν ] if d = 0, and for s ∈ [0 , ) ∩ [0 , ν ] if d >
0, and (cid:107) h sP v (cid:107) γ ≤ C B (cid:107) v (cid:107) ˜ H − s ( γ ) , v ∈ S dP , (26)for s ∈ [0 , ν ], which hold for admissible partitions P ∈ adm( P ), with the constant C B = C B ( σ s , σ g ). Recall that γ = int (cid:83) τ ∈ Q τ for some Q ⊆ P . The both inequalities are provedin Dahmen, Faermann, Graham, Hackbusch, and Sauter (2004) in a more general setting,with a piecewise affine mesh-size function h P . Nevertheless, by local quasi-uniformity, theirresults immediately imply (25) and (26) with piecewise constant h P , since in our setting h P enters only in a derivative-free fashion. Finally, we will make crucial use of inequalitiesof the type (cid:88) τ ∈ P h s + t ) τ (cid:107) Av (cid:107) s,τ ≤ C A (cid:107) v (cid:107) H t (Γ) , v ∈ S dP , (27)that is assumed to hold for admissible partitions P ∈ adm( P ), with C A = C A ( A, σ s , σ g ).This inequality is somewhat more demanding than the standard inverse estimates sinceit involves the non-local operator A , and in some sense it requires A to be almost local.In fact, boundary integral operators have certain locality properties, which is exploited inour analysis only through this inequality. We prove it in Theorem 6.1 for a wide classof boundary integral operators, but in general allowing only C , surfaces. Feischl et al.(2011a,b) prove (27) for s = 1, and A equal to the simple layer potential operator onpolyhedral surfaces. In this section, we focus on the case where the operator A is of order zero, i.e., the case t = 0. This is a nice model case to test our arguments on. On a practical side, this caseis a representative of Hilbert-Schmidt operators on general domains or manifolds, and a model of boundary integral operators associated to the double layer potential. In the lattercase, for instance when the surface Γ is C so that the double layer potential operator iscompact, one only has a G˚arding-type inequality instead of the strict coercivity (7), but Operators of order zero 11 we expect that such cases can be handled at the expense of requiring a sufficiently fineinitial mesh, in the spirit of Mekchay and Nochetto (2005) and Gantumur (2008).The surface Γ can be either closed or open. We employ the piecewise constants S P .The Galerkin approximation of u from S P is denoted by u P ∈ S P , and the correspondingresidual — by r P . Recall the notations (cid:107) · (cid:107) and (cid:107) · (cid:107) ω for the L -norms on Γ and ω ⊂ Γ,respectively.The equivalence (13) provides the convenient starting point α (cid:107) u − u P (cid:107) ≤ (cid:107) r P (cid:107) ≤ β (cid:107) u − u P (cid:107) , (28)which suggests us to use local L -norms of the residual as error indicators. Below we provea localized version of (28) with the error replaced by the difference between two Galerkinapproximations. The simple observations in its proof are the essence of this paper, in thesense that the rest of the paper can be thought of as an attempt to exploit their naturalconsequences and to extend the arguments to non-zero order operators. Note that noexplicit condition whatsoever is imposed on the locality of A . Lemma 3.1.
Let
P, P (cid:48) ∈ [ P ] be partitions with P (cid:22) P (cid:48) , and let Γ ∗ = int (cid:83) τ ∈ P \ P (cid:48) τ . Thenwe have α (cid:107) u P − u P (cid:48) (cid:107) ≤ (cid:107) r P (cid:107) Γ ∗ ≤ β (cid:107) u P − u P (cid:48) (cid:107) + 2 (cid:107) r P − v (cid:107) Γ ∗ , (29) for any function v ∈ S P (cid:48) .Proof. Recall that “int” denotes the interior, and that P \ P (cid:48) = { τ ∈ P : τ (cid:54)∈ P (cid:48) } , so Γ ∗ is the region covered by the refined triangles. Let e = u P (cid:48) − u P , and let e P ∈ S P be the L -orthogonal projection of e onto S P . Then from the Galerkin condition (10) we get (cid:104) Ae, e (cid:105) = (cid:104) r P , e (cid:105) = (cid:104) r P , e − e P (cid:105) ≤ (cid:107) r P (cid:107) Γ ∗ (cid:107) e − e P (cid:107) Γ ∗ ≤ (cid:107) r P (cid:107) Γ ∗ (cid:107) e (cid:107) Γ ∗ , (30)where we have used that e = e P outside Γ ∗ . This proves the first inequality in (29).To prove the second inequality, let v ∈ S P (cid:48) be supported in Γ ∗ . Then we have (cid:107) v (cid:107) ∗ = (cid:104) v, v (cid:105) = (cid:104) v − r P , v (cid:105) + (cid:104) A ( u P (cid:48) − u P ) , v (cid:105)≤ ( (cid:107) v − r P (cid:107) Γ ∗ + (cid:107) A ( u P (cid:48) − u P ) (cid:107) Γ ∗ ) (cid:107) v (cid:107) Γ ∗ , (31)implying that (cid:107) r P (cid:107) Γ ∗ ≤ (cid:107) r P − v (cid:107) Γ ∗ + (cid:107) v (cid:107) Γ ∗ ≤ (cid:107) r P − v (cid:107) Γ ∗ + (cid:107) A ( u P (cid:48) − u P ) (cid:107) , (32)which gives the second inequality in (29). Remark . The arguments used in the preceding proof are of course inspired by thecorresponding finite element theory. However, especially the argument (31) for the lowerbound seems to have a new flavour, in that it does not break the action of A up into element-wise (or star-wise) operations, making it particularly suitable for nonlocal operators. Operators of order zero 12
We recognize the term (cid:107) r P − v (cid:107) Γ ∗ in (29) as an oscillation term, which measures howmuch of the residual is captured when we move from P to P (cid:48) . It is of interest to controlthis term. Since S P ⊂ H r (Γ) for r ∈ (0 , ), assuming that A : H r (Γ) → H r (Γ) is boundedfor all r ∈ (0 , ), we have Au P ∈ H r (Γ) for r in the same range. Now, assuming that f ∈ H r (Γ , P ) for some r ∈ (0 , ), this ensures r P ∈ H r (Γ , P ), therefore from (21) we haveinf v ∈ S P (cid:48) (cid:107) r P − v (cid:107) ∗ ≤ C J (cid:88) τ ∈ P \ P (cid:48) h rτ | r P | r,τ . (33)This suggests us to define the residual oscillation osc r ( v, P, ω ) := (cid:88) τ ∈ P, τ ⊂ ω h rτ | f − Av | r,τ , (34)for ω ⊆ Γ and v ∈ S P , so that (29) implies α (cid:107) u P − u P (cid:48) (cid:107) ≤ (cid:107) r P (cid:107) Γ ∗ ≤ β (cid:107) u P − u P (cid:48) (cid:107) + 2 C J osc r ( u P , P, Γ ∗ ) . (35)If the oscillation term is sufficiently small, the difference between two discrete solutions iscompletely controlled by a local L -norm of the residual. In this sense, the first inequalityin (35) is an example of a local discrete upper bound , while the second one is that of a local discrete lower bound . These bounds are also called local discrete reliability and localdiscrete efficiency , respectively.In the following, we will fix some r ∈ (0 , that f ∈ H r (Γ , P ). Notethat this is in the same spirit as assuming f ∈ L in the context of second order ellipticequations, even though there the weak formulation is well-posed for f ∈ H − . We willuse the convenient abbreviation osc r ( P, ω ) = osc r ( u P , P, ω ). The next lemma collectscrucial properties of the oscillation osc r ( P, Γ) and the combination ||| u − u P ||| + osc r ( P, Γ) , assuming an inverse-type inequality, which shall be verified in §
6. As it turns out, thisinverse-type inequality is also sufficient to guarantee the finiteness of oscillation (34). Thequantity ||| u − u P ||| + osc r ( P, Γ) , the counterpart of the total error in Cascon, Kreuzer,Nochetto, and Siebert (2008), will be the main character in our subsequent analysis. Lemma 3.3.
Assume that (cid:88) τ ∈ P h rτ | Av | r,τ ≤ C A ||| v ||| , v ∈ S P , (36) for P ∈ adm( P ) , with the constant C A = C A ( A, σ s , σ g ) . Then the oscillation (34) is finitefor any v ∈ S P and P ∈ adm( P ) . Moreover, the followings hold. In view of (34), one has a computational advantage if r = 1, since there would be no fractional norms. Operators of order zero 13 a) There is a constant C G > such that ||| u − u P ||| + osc r ( u P , P, Γ) ≤ C G inf v ∈ S P (cid:0) ||| u − v ||| + osc r ( v, P, Γ) (cid:1) , (37) for any P ∈ adm( P ) .b) There exists a constant λ > such that osc r ( P (cid:48) , Γ) ≤ (1 + δ )osc r ( P, Γ) − λ (1 + δ )osc r ( P, Γ ∗ ) + C δ ||| u P − u P (cid:48) ||| , (38) for any P, P (cid:48) ∈ adm( P ) with P (cid:22) P (cid:48) , and for any δ > , with C δ depending on δ ,where Γ ∗ = int (cid:83) τ ∈ P \ P (cid:48) τ .c) Let P, P (cid:48) ∈ adm( P ) be such that P (cid:22) P (cid:48) , and let Γ ∗ = int (cid:83) τ ∈ P (cid:48) \ P τ . If, for some µ ∈ (0 , ) it holds that ||| u − u P (cid:48) ||| + osc r ( P (cid:48) , Γ) ≤ µ (cid:0) ||| u − u P ||| + osc r ( P, Γ) (cid:1) , (39) then with θ = α (1 − µ ) β (1+2 C A ) , we have (cid:107) r P (cid:107) ∗ + osc r ( P, Γ ∗ ) ≥ θ (cid:0) (cid:107) r P (cid:107) + osc r ( P, Γ) (cid:1) . (40) Remark . The estimate (36) is proved in Section 6 for a general class of singular integraloperators of order zero, under the hypothesis that ˆ A : H σ (Ω) → H σ (Ω) is bounded forsome σ > n and that r ∈ (0 , σ ), where ˆ A is an extension of A to Ω if Γ is open, andˆ A = A if Γ = Ω. Recalling that Ω is a C ν − , -manifold, the condition on σ translates to ν > n , see Remark 6.2 for details. Therefore Lipschitz curves are allowed, but for n = 2we need a C , surface. Note that there is no condition on the regularity of the boundaryof Γ other than the Lipschitz condition, when Γ is an open surface. In any case, we believethat the restriction ν > n is an artifact of our proof, and anticipating future weakeningsof this restriction, the rest of this section will be presented so that it depends only on theassumption (36). Proof of Lemma 3.3.
Let w ∈ S P satisfy ||| u − w ||| + osc r ( w, P, Γ) ≤ v ∈ S P (cid:0) ||| u − v ||| + osc r ( v, P, Γ) (cid:1) . (41)Then from the definition of oscillation, we haveosc r ( u P , P, Γ) ≤ (cid:88) τ ∈ P h rτ | f − Aw | r,τ + 2 (cid:88) τ ∈ P h rτ | A ( w − u P ) | r,τ ≤ r ( w, P, Γ) + 2 C A ||| w − u P ||| , (42) Operators of order zero 14 where in the last step we have used inverse inequality (36). This gives ||| u − u P ||| + osc r ( u P , P, Γ) ≤ ||| u − u P ||| + 2 C A ||| w − u P ||| + 2 osc r ( w, P, Γ) , (43)and upon using the Galerkin orthogonality ||| u − u P ||| + ||| w − u P ||| = ||| u − w ||| , we obtain(37) with, say C G = 4(1 + C A ).Now we turn to b). With λ ∈ (0 ,
1) being the contraction factor for h τ when τ isrefined once, we have (cid:88) τ ∈ P (cid:48) \ P h rτ | r P | r,τ ≤ (cid:88) τ ∈ P \ P (cid:48) λ r h rτ | r P | r,τ = λ r osc r ( P (cid:48) , Γ ∗ ) . (44)Using this inosc r ( P (cid:48) , Γ) ≤ (1 + δ ) (cid:88) τ ∈ P (cid:48) h rτ | r P | r,τ + (1 + 1 δ ) (cid:88) τ ∈ P (cid:48) h rτ | A ( u P (cid:48) − u P ) | r,τ , (45)and by using the inverse inequality (36) on the last term, we establish (38).For c), from (39) we infer(1 − µ )( ||| u − u P ||| + osc r ( P, Γ) ) ≤ ||| u P − u P (cid:48) ||| + osc r ( P, Γ) − r ( P (cid:48) , Γ) ≤ (1 + 2 C A ) ||| u P − u P (cid:48) ||| + osc r ( P, Γ ∗ ) ≤ β (1 + 2 C A ) α (cid:107) r P (cid:107) Γ ∗ + osc r ( P, Γ ∗ ) , (46)where we have used the Galerkin orthogonality, the estimateosc r ( P, Γ \ Γ ∗ ) ≤ r ( P (cid:48) , Γ \ Γ ∗ ) + 2 C A ||| u P − u P (cid:48) ||| , (47)the local discrete upper bound in (35), and the norm equivalence (8). The proof is com-pleted upon noting that (cid:107) r P (cid:107) ≤ γ ||| u − u P ||| , (48)where γ = β /α ≥
1, which is a combination of the global lower bound in (13), and thenorm equivalence (8).Once we have the preceding results, and given the techniques developed in Stevenson(2007) and Cascon et al. (2008), it is a fairly straightforward matter to obtain geometricerror reduction and quasi-optimality for an adaptive method such as the ones consideredhere. Nevertheless, we include detailed proofs for convenience of the reader. Our first stopis the contraction property of the adaptive method.
Operators of order zero 15
Proposition 3.5.
Let the assumption (36) of Lemma 3.3 hold. Let
P, P (cid:48) ∈ adm( P ) beadmissible partitions with P (cid:22) P (cid:48) , and let Γ ∗ = int (cid:83) τ ∈ P \ P (cid:48) τ . Suppose, for some θ ∈ (0 , that (cid:107) r P (cid:107) ∗ + osc r ( P, Γ ∗ ) ≥ θ (cid:0) (cid:107) r P (cid:107) + osc r ( P, Γ) (cid:1) . (49) Then there exist constants γ ≥ and µ ∈ (0 , such that ||| u − u P (cid:48) ||| + γ osc r ( P (cid:48) , Γ) ≤ µ (cid:0) ||| u − u P ||| + γ osc r ( P, Γ) (cid:1) . (50) Proof.
The property (49), together with the global upper bound in (13), the local discretelower bound in (35), and the norm equivalence (8), gives θ (cid:18) α β ||| u − u P ||| + osc r ( P, Γ) (cid:19) ≤ β α ||| u P − u P (cid:48) ||| + 2(2 C J + 1) osc r ( P, Γ ∗ ) . (51)Then for any γ ≥
0, we combine Lemma 3.3 b) and (51) to infer ||| u − u P (cid:48) ||| + γ osc r ( P (cid:48) , Γ) = ||| u − u P ||| − ||| u P − u P (cid:48) ||| + γ osc r ( P (cid:48) , Γ) ≤ ||| u − u P ||| − (1 − γC δ ) ||| u P − u P (cid:48) ||| + γ (1 + δ )osc r ( P, Γ) − γλ (1 + δ )osc r ( P, Γ ∗ ) ≤ (cid:18) − θα (1 − γC δ )2 β (cid:19) ||| u − u P ||| + (cid:18) γ (1 + δ ) − θα (1 − γC δ )2 β (cid:19) osc r ( P, Γ) + (cid:18) α (2 C J + 1) (1 − γC δ ) β − γλ (1 + δ ) (cid:19) osc r ( P, Γ ∗ ) =: µ ||| u − u P ||| + µ γ osc r ( P, Γ) + µ osc r ( P, Γ ∗ ) . (52)We choose γ , depending on δ >
0, so that µ = 0, i.e., so that1 γ = C δ + λβ (1 + δ ) α (2 C J + 1) . (53)With this choice and for δ > µ = 1 − α γλθ (1 + δ )2(2 C J + 1) < , (54)and µ = (1 + δ ) (cid:18) − λθ C J + 1) (cid:19) < , (55)establishing the proof.To be explicit, the importance of the preceding proposition is the following. Supposethat we have an admissible partition P and the discrete solution u P ∈ S P . Then the localresidual norm (cid:107) r P (cid:107) τ and the local oscillation osc r ( P, τ ) are computable for τ ∈ P , and Operators of order zero 16 by selecting a set R ⊂ P of triangles such that Γ ∗ = int (cid:83) τ ∈ R τ satisfies (49) for some θ ∈ (0 , P (cid:48) of P such that P \ P (cid:48) ⊇ R , i.e.,that each triangle in R is refined at least once, we can guarantee e ( P (cid:48) ) ≤ µe ( P ) for some µ <
1, where e ( Q ) = ||| u − u Q ||| + γ osc r ( Q, Γ) . By repeatedly applying this procedure,we can ensure convergence ||| u − u P ||| → P runs over the partitions generated by thealgorithm. This however, does not say anything about the growth of P , a fundamentalquestion that will be addressed below.For u ∈ L (Γ) the solution of Au = f with f ∈ H r (Γ , P ), and for P ∈ [ P ], we definedist r ( u, S P ) = min v ∈ S P (cid:0) ||| u − v ||| + osc r ( v, P, Γ) (cid:1) , (56)where the implicit dependence of oscillation on r has been made explicit. Note that theminimum exists since S P is finite dimensional. Furthermore, for ε > r ( u, ε ) = min { P − P : P ∈ adm( P ) , dist r ( u, S P ) ≤ ε } . (57)Hence card r ( u, ε ) is in certain sense the cardinality of a smallest admissible partition P that is able to support a function that is within an ε distance from u . Note that card r ( u, ε )is finite for any ε >
0, since from the discussion in the preceding paragraph, there is asequence of (finite) partitions { P k } ⊂ adm( P ) with dist r ( u, S P k ) → D¨orfler’s marking strategy in literature.
Proposition 3.6.
Let the assumption (36) of Lemma 3.3 hold. Let P ∈ adm( P ) , andlet θ ∈ (0 , θ ∗ ) with θ ∗ = α β (1+2 C A ) . Suppose that R ⊆ P is a subset whose cardinality isminimal up to a constant factor κ ≥ , among all R ⊆ P satisfying (cid:107) r P (cid:107) ∗ ( R ) + osc r ( P, Γ ∗ ( R )) ≥ θ (cid:0) (cid:107) r P (cid:107) + osc r ( P, Γ) (cid:1) . (58) where Γ ∗ ( R ) = int (cid:83) τ ∈ R τ . Then we have R ≤ κ card r ( u, ε ) , (59) where ε is defined by ε = θ ∗ − θ C G θ ∗ (cid:0) ||| u − u P ||| + osc r ( P, Γ) (cid:1) . (60) Proof.
Let us introduce the abbreviation e ( Q ) = ||| u − u Q ||| + osc( Q, Γ) , Q ∈ [ P ] , (61)and let P ε ∈ adm( P ) be such that P ε − P ≤ card r ( u, ε ) , and e ( P ε ) ≤ ε . (62) Operators of order zero 17
Then for ˜ P = P ⊕ P ε , from Lemma 3.3 a) we have e ( ˜ P ) ≤ C G e ( P ε ) ≤ C G ε = µe ( P ) , (63)where µ = θ ∗ − θ θ ∗ ∈ (0 , ), and so an application of Lemma 3.3 c) gives (cid:107) r P (cid:107) ∗ ( P \ ˜ P ) + osc r ( P, Γ ∗ ( P \ ˜ P )) ≥ θ (cid:0) (cid:107) r P (cid:107) + osc r ( P, Γ) (cid:1) . (64)Recalling that R minimizes R up to the constant factor κ among all subsets P \ ˜ P satisfying the preceding inequality, we infer that R ≤ κ P \ ˜ P ), and taking into accountthat P \ ˜ P ) ≤ P − P and the estimate in (17), we get R ≤ κ ( P − P ) ≤ κ ( P ε − P ) ≤ κ card r ( u, ε ) , (65)completing the proof.We have almost all the ingredients to give a bound on the growth of P , and to discusswhether this growth rate is optimal in one or another sense. Let us start by making theterm “adaptive BEM” precise. Algorithm 1:
Adaptive BEM ( t = 0) parameters : conforming partition P , and θ ∈ [0 , output : P k ∈ adm( P ) and u k ∈ S P k for all k ∈ N for k = 0 , , . . . do Compute u k ∈ S P k as the Galerkin approximation of u from S P k ; Identify a minimal (up to a constant factor) set R k ⊂ P k of triangles satisfying (cid:107) r k (cid:107) ∗ + osc( P k , Γ ∗ ) ≥ θ (cid:0) (cid:107) r k (cid:107) + osc( P k , Γ) (cid:1) , (66)where r k = f − Au k and Γ ∗ = int (cid:83) τ ∈ R k τ ; Set P k +1 = refine( P k , R k ); endfor Now we turn to the issue of convergence rate. Since Proposition 3.6 gives a bound on R k , we simply need to use (16) to get P k − P ≤ κC c k − (cid:88) m =0 card r ( u, Ce ( P m )) , (67)where C is the constant from (60), and e ( P m ) is as in (61). On the other hand, Proposition3.5 guarantees a geometric decrease of e ( P m ), i.e, we have e ( P k ) ≤ Cµ k − m e ( P m ) , hence card r ( u, Ce ( P m )) ≤ card r ( u, Cµ m − k e ( P k )) , (68) Operators of order zero 18 for some constants
C > < µ <
1. Note that C denotes different constants in itsdifferent appearances. Therefore, if our particular u ∈ A − ( H r (Γ , P )) satisfiescard r ( u, λε ) ≤ Cλ − /s card r ( u, ε ) , λ > , ε > , (69)for some constant s >
0, then we would get what can be called instance optimality P k − P ≤ C card r ( u, e ( P k )) . (70)However, it is not clear how to usefully characterize the set of such u , so we settle forsomething more modest. Instead of (69), if we havecard r ( u, ε ) ≤ C u ε − /s , ε > , (71)for some constant s >
0, then we get what can be called class optimality P k − P ≤ CC u e ( P k ) − /s . (72)This motivates us to define the approximation class A r,s ⊂ A − ( H r (Γ , P )) with s ≥ u for which | u | A r,s = sup ε> (card r ( u, ε ) s ε ) < ∞ . (73)Thus A r,s is characterized by card r ( u, ε ) ≤ ε − /s | u | A r,s , (74)or equivalently, bymin { dist r ( u, S P ) : P ∈ adm( P ) , P − P ≤ N } ≤ N − s | u | A r,s . (75)We have proved the following. Theorem 3.7.
Let the assumption (36) of Lemma 3.3 hold, and in Algorithm 1, supposethat θ ∈ (0 , θ ∗ ) with θ ∗ = α β (1+2 C A ) . Let f ∈ H r (Γ , P ) and u ∈ A r,s for some s > . Thenwe have ||| u − u k ||| + osc r ( u k , P k , Γ) ≤ C | u | A r,s ( P k − P ) − s , (76) where C > is a constant. Positive order operators 19
In this section, we consider the case t ∈ [0 , ). We assume that Γ is open, and remarkthat the closed case can be treated with similar methods. Recall that the equation we aredealing with is Au = f with linear homeomorphism A : ˜ H t (Γ) → H − t (Γ). For simplicitywe consider right hand sides satisfying f ∈ L (Γ). We employ the continuous piecewiseaffine functions with homogeneous boundary conditions S P := ˜ S P . If n ≥
2, the admissiblepartitions will be used synonymous with conforming triangulations, with the newest vertexbisection algorithm for refinements. So we have S P ⊂ ˜ H t (Γ), and we make an additionalassumption that A : ˜ H t (Γ) → L (Γ) is bounded, in order to keep the useful property f − Av ∈ L (Γ) for any finite element function v ∈ S P . Note that this assumption issatisfied for our main example – the hypersingular integral operator. Note also that if t > then we need ν ≥
2, i.e., we need the space Ω in which Γ lies to be at least a C , manifold.Starting from this section, we shall often dispense with giving explicit names to con-stants, and use the Vinogradov-style notation X (cid:46) Y , which means X ≤ C · Y with someconstant C that is allowed to depend only on the operator A and on (geometry of) theset of admissible partitions adm( P ). The following is an extension of Lemma 3.1 to t ≥ ω ∗ includesa buffer layer of triangles around the refined triangles, so that ω ∗ = ω (Γ ∗ ) with Γ ∗ asin Lemma 3.1. By using the Scott-Zhang quasi-interpolation operator in the proof, it ispossible to do without the buffer layer if t > , which is however not terribly exciting sincethe relevant case to us is t = .We define the oscillation for v ∈ S P byosc( v, P ) = (cid:107) h tP ( f − Av ) − h − tP w (cid:107) , (77)where, with ˆ P the uniform refinement of P , w = Q ˆ P h tP ( f − Av ) ∈ S ˆ P is the Cl´ementinterpolator of h tP ( f − Av ), given by Q ˆ P g = (cid:88) z ∈ N ˆ P \ ∂ Γ g z ( z ) φ z , (78)where N ˆ P is the set of all nodes in ˆ P , φ z ∈ S ˆ P is the standard nodal basis function at z ,and g z ∈ P is the L -orthogonal projection of g onto the affine functions on the star ˆ ω ( z )around z with respect to ˆ P . Lemma 4.1.
Let
P, P (cid:48) ∈ adm( P ) with P (cid:22) P (cid:48) , and let ω ∗ = (cid:83) τ ∈ P \ P (cid:48) ω ( τ ) . Then wehave ||| u P − u P (cid:48) ||| (cid:46) (cid:107) h tP r P (cid:107) ω ∗ (cid:46) (cid:107) h P /h P (cid:48) (cid:107) t ∞ ||| u P − u P (cid:48) ||| + (cid:107) h tP r P − h − tP v (cid:107) ω ∗ , (79) for any v ∈ S P (cid:48) with supp v ⊆ ω ∗ , where (cid:107) · (cid:107) ∞ denotes the L ∞ -norm. Moreover, we havethe following global bounds ||| u − u P ||| (cid:46) (cid:107) h tP r P (cid:107) (cid:46) (cid:107) h P /h P (cid:48) (cid:107) t ∞ ||| u − u P ||| + (cid:107) h tP r P − h − tP v (cid:107) , (80) Positive order operators 20 for any v ∈ S P (cid:48) .Furthermore, for γ ⊆ Γ and ˆ ω ( γ ) = (cid:83) { ˆ ω ( z ) : z ∈ N ˆ P ∩ γ } , we have (cid:107) h tP r P − h − tP Q ˆ P h tP r P (cid:107) γ ≤ C Q (cid:107) h tP r P (cid:107) ω ( γ ) , (81) for some constant C Q > , i.e., the estimator dominates the oscillation on the Galerkinsolutions. In particular, we have the equivalence α (cid:107) h tP r P (cid:107) ≤ ||| u − u P ||| + osc( u P , P ) (cid:46) (cid:107) h tP r P (cid:107) , (82) where α > is a contstant.Remark . The global upper bound (i.e., the first inequality) of (80) has been establishedin Carstensen, Maischak, Praetorius, and Stephan (2004), and a similar bound involvinglocal L p -norms on stars appears in Nochetto, von Petersdorff, and Zhang (2010). Proof of Lemma 4.1.
For v ∈ ˜ H t (Γ) and v P = Q P v ∈ S P being the quasi-interpolant of v as in (24), we have (cid:104) r P , v (cid:105) = (cid:104) r P , v − v P (cid:105) = (cid:104) h tP r P , h − tP ( v − v P ) (cid:105)≤ (cid:107) h tP r P (cid:107)(cid:107) h − tP ( v − v P ) (cid:107) (cid:46) (cid:107) h tP r P (cid:107)(cid:107) v (cid:107) ˜ H t (Γ) , (83)where in the last step we have used (cid:88) τ ∈ P h − tτ (cid:107) v − v P (cid:107) τ (cid:46) (cid:88) τ ∈ P (cid:107) v (cid:107) H t ( ω ( τ )) (cid:46) (cid:107) v (cid:107) H t (Γ) , (84)by local finiteness. This gives the first inequality in (80).With v = u P (cid:48) − u P and v P = Q P v ∈ S P , we infer (cid:104) Av, v (cid:105) = (cid:104) r P , v − v P (cid:105) = (cid:104) r P , v − v P (cid:105) ω ∗ ≤ (cid:107) h tP r P (cid:107) ω ∗ (cid:107) h − tP ( v − v P ) (cid:107) ω ∗ (cid:46) (cid:107) h tP r P (cid:107) ω ∗ (cid:107) v (cid:107) ˜ H t (Γ) , (85)where we have used the fact that v P = v outside ω ∗ . This gives the first inequality in (79).Let either e = u P (cid:48) − u P or e = u − u P . Then in both cases, for v ∈ S P (cid:48) and ω ⊇ supp v ,we have (cid:104) h − tP v, h − tP v (cid:105) ω = (cid:104) h − tP v, v (cid:105) = (cid:104) h − tP v − r P , v (cid:105) + (cid:104) Ae, v (cid:105) = (cid:104) h − tP v − h tP r P , h − tP v (cid:105) + (cid:104) Ae, v (cid:105)≤ (cid:107) h − tP v − h tP r P (cid:107) ω (cid:107) h − tP v (cid:107) ω + (cid:107) Ae (cid:107) H − t (Γ) (cid:107) v (cid:107) ˜ H t (Γ) (cid:46) (cid:107) h − tP v − h tP r P (cid:107) ω (cid:107) h − tP v (cid:107) ω + (cid:107) Ae (cid:107) H − t (Γ) (cid:107) h − tP (cid:48) v (cid:107) ω , (86)so that (cid:107) h − tP v (cid:107) ω (cid:46) (cid:107) h − tP v − h tP r P (cid:107) ω + (cid:107) h P /h P (cid:48) (cid:107) t ∞ (cid:107) e (cid:107) ˜ H t (Γ) . (87) Positive order operators 21
From this, we infer (cid:107) h tP r P (cid:107) ω ≤ (cid:107) h tP r P − h − tP v (cid:107) ω + (cid:107) h − tP v (cid:107) ω (cid:46) (cid:107) h tP r P − h − tP v (cid:107) ω + (cid:107) h P /h P (cid:48) (cid:107) t ∞ (cid:107) e (cid:107) ˜ H t (Γ) , (88)proving the both second inequalities in (79) and (80). Remark . For a large number of non-residual a posteriori errorestimators for the hypersingular integral equation on curves, Erath, Funken, Goldenits, andPraetorius (2009b) proved that the estimators are equivalent to the global error, with theupper bound depending on the saturation assumption: ||| u − u P ||| (cid:46) ||| ˆ u P − u P ||| , where ˆ u P is the Galerkin approximation from some enriched space ˆ S P ⊃ S P , which is typically thepiecewise linears on the uniform refinement of P . Combining the discrete lower bound withthe global upper bound from Lemma 4.1, we have ||| u − u P ||| (cid:46) ||| u P − u P (cid:48) ||| + (cid:107) h tP r P − h − tP v (cid:107) , (89)for any v ∈ S P (cid:48) , where P (cid:48) is the uniform refinement of P . At least in theory, this confirmsthe saturation assumption up to an oscillation term. In practice though, to control theoscillation as defined here, it seems that the residual needs to be computed anyways. Thequestion of whether such an overhead is tolerable calls for further investigation.Let us get back to the residual based error indicators (cid:107) h tτ r P (cid:107) τ from Lemma 4.1. Inview of the results in that lemma, the circumstances are very similar to what happens inthe finite element case, and in particular, the local quantities (cid:107) h tP r P (cid:107) τ as error indicatorswill give rise to an adaptive algorithm that converges quasi-optimally in a certain sense. Lemma 4.4.
Assume that (cid:107) h tP Av (cid:107) ≤ C A ||| v ||| , v ∈ S P , (90) for P ∈ adm( P ) , with the constant C A = C A ( A, σ s , σ g ) . Then the followings hold.a) There is a constant C G > such that ||| u − u P ||| + osc( u P , P ) ≤ C G inf v ∈ S P (cid:0) ||| u − v ||| + osc( v, P ) (cid:1) , (91) for any P ∈ adm( P ) .b) There exists a constant λ > such that (cid:107) h tP (cid:48) r P (cid:48) (cid:107) ≤ (1 + δ ) (cid:107) h tP r P (cid:107) − λ (1 + δ ) (cid:107) h tP r P (cid:107) ∗ + C δ ||| u P − u P (cid:48) ||| , (92) for any P, P (cid:48) ∈ adm( P ) with P (cid:22) P (cid:48) , and for any δ > , with C δ depending on δ ,where Γ ∗ = int (cid:83) τ ∈ P \ P (cid:48) τ . Positive order operators 22 c) Let
P, P (cid:48) ∈ adm( P ) be such that P (cid:22) P (cid:48) , and let Γ ∗ = int (cid:83) τ ∈ P \ P (cid:48) τ . If, for some µ ∈ (0 , ) it holds that ||| u − u P (cid:48) ||| + osc( u P (cid:48) , P (cid:48) ) ≤ µ (cid:0) ||| u − u P ||| + osc( u P , P ) (cid:1) , (93) then with θ = α (1 − µ ) C Q + β (1+2 C A C Q ) and ω ∗ = (cid:83) τ ∈ P \ P (cid:48) ω ( τ ) , we have (cid:107) h tP r P (cid:107) ω ∗ ≥ θ (cid:107) h tP r P (cid:107) , (94) where β > is a constant implicit in the local discrete upper bound in (79) , such that ||| u P − u P (cid:48) ||| ≤ β (cid:107) h tP r P (cid:107) ω ∗ .Remark . The estimate (90) is proved in Section 6 for a general class of singular integraloperators of positive order, under the hypothesis that ˆ A : H t + σ (Ω) → H − t + σ (Ω) is boundedfor some σ > max { n , t } , where ˆ A is an extension of A to Ω. Recalling that Ω is a C ν − , -manifold, and assuming that t ≤ n , the condition on σ translates to ν > n + t , seeRemark 6.2 for details. Therefore for n = 1 and n = 2 we need a C , curve or a surface,respectively, assuming that ≤ t <
1. Note that the boundary of Γ is allowed to beLipschitz. Anticipating future developments on (90), the rest of this section is presentedso as to depend only on the assumption (90).
Proof of Lemma 4.4.
For v ∈ S P we haveosc( u P , P ) = (cid:107) h tP r P − h − tP Q ˆ P h tP r P (cid:107)≤ osc( v, P ) + (cid:107) h tP A ( v − u P ) − h − tP Q ˆ P h tP A ( v − u P ) (cid:107)≤ osc( v, P ) + C / Q (cid:107) h tP A ( v − u P ) (cid:107) ≤ osc( v, P ) + C / Q C / A ||| v − u P ||| . (95)With this estimate at hand, part a) of the lemma can be proven in exactly the same wayas Lemma 3.3 a).Part b) follows from (cid:107) h tP (cid:48) r P (cid:48) (cid:107) ≤ (1 + δ ) (cid:107) h tP (cid:48) r P (cid:107) + C A (1 + δ − ) ||| u P − u P (cid:48) ||| , (96)and the fact that h P (cid:48) ≤ λ h P on Γ ∗ for some constant λ < α (1 − µ ) (cid:107) h tP r P (cid:107) ≤ (1 − µ ) (cid:0) ||| u − u P ||| + osc( u P , P ) (cid:1) ≤ ||| u − u P ||| + osc( u P , P ) − ||| u − u P (cid:48) ||| − u P (cid:48) , P (cid:48) ) ≤ ||| u P − u P (cid:48) ||| + osc( u P , P ) − u P (cid:48) , P (cid:48) ) . (97)For τ ∈ ˆ P ∩ ˆ P (cid:48) with all its neighbors also in ˆ P ∩ ˆ P (cid:48) , where ˆ P (cid:48) is the uniform refinementof P (cid:48) , we have ( Q ˆ P v ) | τ = ( Q ˆ P (cid:48) v ) | τ , and also h P = h P (cid:48) there. Hence, with ˆ ω = (cid:83) { τ ∈ ˆ P : Positive order operators 23 ∃ σ ∈ P \ P (cid:48) , τ ∩ σ (cid:54) = ∅ } , we have (cid:107) h tP r P − h − tP Q ˆ P h tP r P (cid:107) Γ \ ˆ ω ≤ (cid:107) h tP (cid:48) r P − h − tP (cid:48) Q ˆ P (cid:48) h tP (cid:48) r P (cid:107)≤ (cid:107) h tP (cid:48) r P (cid:48) − h − tP (cid:48) Q ˆ P (cid:48) h tP (cid:48) r P (cid:48) (cid:107) + (cid:107) h tP (cid:48) A ( u P − u P (cid:48) ) − h − tP (cid:48) Q ˆ P h tP (cid:48) A ( u P − u P (cid:48) ) (cid:107)≤ osc( u P (cid:48) , P (cid:48) ) + C / Q (cid:107) h tP (cid:48) A ( u P − u P (cid:48) ) (cid:107) . (98)Combining this with (81), we inferosc( u P , P ) ≤ u P (cid:48) , P (cid:48) ) + C Q (cid:107) h tP r P (cid:107) ω ∗ + 2 C Q C A ||| u P − u P (cid:48) ||| , (99)which, then on account of (97) and the local discrete upper bound, proves the claim.Now we prove an analogue of Proposition 3.5 on error reduction. Proposition 4.6.
Let the assumption (90) of Lemma 4.4 hold. Let
P, P (cid:48) ∈ adm( P ) bewith P (cid:22) P (cid:48) , and let Γ ∗ = int (cid:83) τ ∈ P \ P (cid:48) τ . Suppose, for some ϑ ∈ (0 , that (cid:107) h tP r P (cid:107) Γ ∗ ≥ ϑ (cid:107) h tP r P (cid:107) . (100) Then there exist constants γ ≥ and µ ∈ (0 , such that ||| u − u P (cid:48) ||| + γ (cid:107) h tP (cid:48) r P (cid:48) (cid:107) ≤ µ (cid:0) ||| u − u P ||| + γ (cid:107) h tP r P (cid:107) (cid:1) . (101) Proof.
From the Galerkin orthogonality and Lemma 4.4 b) we have ||| u − u P (cid:48) ||| + γ (cid:107) h tP (cid:48) r P (cid:48) (cid:107) = ||| u − u P ||| − ||| u P − u P (cid:48) ||| + γ (cid:107) h tP (cid:48) r P (cid:48) (cid:107) ≤ ||| u − u P ||| + γ (1 + δ ) (cid:107) h tP r P (cid:107) − γλ (1 + δ ) (cid:107) h tP r P (cid:107) ∗ + ( γC δ − ||| u P − u P (cid:48) ||| ≤ ||| u − u P ||| + γ (1 + δ ) (cid:107) h tP r P (cid:107) − γλ (1 + δ ) (cid:107) h tP r P (cid:107) ∗ , (102)for 0 < γ ≤ /C δ . The idea, introduced in Cascon, Kreuzer, Nochetto, and Siebert (2008),is to use the global upper bound and the D¨orfler property (100) to bound fractions of thefirst two terms by the third term. For any a, b > ||| u − u P (cid:48) ||| + γ (cid:107) h tP (cid:48) r P (cid:48) (cid:107) ≤ (1 − γa ) ||| u − u P ||| + γ ( Ca + 1 + δ ) (cid:107) h tP r P (cid:107) − γλ (1 + δ ) (cid:107) h tP r P (cid:107) ∗ ≤ (1 − γa ) ||| u − u P ||| + γ ( Ca + 1 + δ − b ) (cid:107) h tP r P (cid:107) + γ (cid:18) bϑ − λ (1 + δ ) (cid:19) (cid:107) h tP r P (cid:107) ∗ , (103)with the constant C coming from the global upper bound. We choose b = λϑ > δ > a > δ > Ca + δ < b . Positive order operators 24
For u ∈ ˜ H t (Γ) the solution of Au = f with f ∈ L (Γ), and for P ∈ adm( P ), we definedist ( u, S P ) = inf v ∈ S P (cid:0) ||| u − v ||| + osc( v, P ) (cid:1) . (104)Furthermore, for ε > ( u, ε ) = min { P − P : P ∈ adm( P ) , dist ( u, S P ) ≤ ε } . (105)The following is an analogue of Proposition 3.6, and we omit the proof, since the proof ofProposition 3.6 can be applied here mutatis mutandis . Proposition 4.7.
Let the assumption (90) of Lemma 4.4 hold. Let P ∈ adm( P ) , and let θ ∈ (0 , θ ∗ ) with θ ∗ = α C Q + β (1+2 C A C Q ) . Suppose that R ⊆ P is a subset whose cardinality isminimal up to a constant factor, among all R ⊆ P satisfying (cid:107) h tP r P (cid:107) ∗ ( R ) ≥ θ (cid:107) h tP r P (cid:107) , (106) where Γ ∗ ( R ) = int (cid:83) τ ∈ R τ . Then we have R (cid:46) card ( u, ε ) , (107) where ε is defined by ε = θ ∗ − θ C G θ ∗ (cid:0) ||| u − u P ||| + osc( P, Γ) (cid:1) . (108)For completeness, in the rest of this section we give an explicit pseudocode for theadaptive BEM, then define the relevant approximation classes, and finally record a theoremon quasi-optimality. Algorithm 2:
Adaptive BEM ( t > parameters : conforming partition P , and θ ∈ [0 , output : P k ∈ adm( P ) and u k ∈ S P k for all k ∈ N for k = 0 , , . . . do Compute u k ∈ S P k as the Galerkin approximation of u from S P k ; Identify a minimal (up to a constant factor) set R k ⊂ P k of triangles satisfying (cid:107) h tP k r k (cid:107) Γ ∗ ≥ θ (cid:107) h tP k r k (cid:107) , (109)where r k = f − Au k and Γ ∗ = int (cid:83) τ ∈ R k τ ; Set P k +1 = refine( P k , R k ); endfor We define the approximation class A ,s ⊂ A − ( L (Γ)) with s ≥ u forwhich | u | A ,s = sup ε> (card ( u, ε ) s ε ) < ∞ . (110)The following result is immediate. Negative order operators 25
Theorem 4.8.
Let the assumption (90) of Lemma 4.4 hold, and in Algorithm 2, supposethat θ ∈ (0 , θ ∗ ) with θ ∗ = α C Q + β (1+2 C A C Q ) . Let f ∈ L (Γ) and u ∈ A ,s for some s > .Then we have ||| u − u k ||| + osc( u k , P k ) ≤ C | u | A ,s ( P k − P ) − s , (111) where C > is a constant. Finally, we turn to the case t <
0. The domain Γ can be either a closed manifold ora connected polygonal subset of a closed manifold. Recall that we are dealing with thelinear homeomorphism A : ˜ H t (Γ) → H − t (Γ). We will use the piecewise constant finiteelement spaces S P , and if n ≥
2, take conforming triangulations as the class of admissiblepartitions. Recall that u P ∈ S P is the Galerkin approximation of u from S P , and that r P = f − Au P is its residual.Faermann (2000, 2002) established the equivalence ||| u − u P ||| (cid:46) (cid:88) z ∈ N P | r P | − t,ω ( z ) (cid:46) ||| u − u P ||| , (112)where N P is the set of all vertices in the triangulation P , and proposed to use the localquantities | r P | − t,ω ( z ) as error indicators for adaptive refinements. An alternative proof, us-ing interpolation spaces appeared in Carstensen, Maischak, and Stephan (2001). However,the discrete local counterparts to this equivalence have been open. On the other hand,the weighted residual type error indicators, h tτ | r P | ,τ for τ ∈ P , have been around for awhile, with a guaranteed global upper bound ||| u − u P ||| (cid:46) (cid:88) τ ∈ P h t ) τ | r P | ,τ , (113)cf. Carstensen et al. (2001). These indicators are computationally more attractive, butfor locally refined meshes no global lower bound or discrete local estimates have beenknown, until the appearance of Feischl, Karkulik, Melenk, and Praetorius (2011a,b) andthis work. The following lemma establishes the missing bounds for the afore-mentionederror indicators. Lemma 5.1.
Let
P, P (cid:48) ∈ adm( P ) be with P (cid:22) P (cid:48) , and let Γ ∗ = int (cid:83) τ ∈ P \ P (cid:48) τ . Further-more, with ω ∗ = ω (Γ ∗ ) and φ z ∈ S P the standard nodal basis function at z ∈ N P , let φ = (cid:80) z ∈ N P ∩ ω ∗ φ z . Then for any v ∈ S P (cid:48) , it holds that ||| u P − u P (cid:48) ||| (cid:46) (cid:88) z ∈ N P ∩ ω ∗ | r P | − t,ω ( z ) + (cid:107) h tP (cid:48) ( r P − v ) (cid:107) ∗ + (cid:107) φ r P − v (cid:107) H − t (Γ) . (114) Negative order operators 26
In addition to the above, for some r ∈ [ − t, ) ∩ (0 , − t ) ∩ [0 , ν ] , let f ∈ H r (Γ) and let A : ˜ H r +2 t (Γ) → H r (Γ) be bounded. Then for any v ∈ S P and w ∈ S P (cid:48) , where ˆ P is theuniform refinement of P , we have (cid:88) z ∈ N P ∩ ω ∗ | r P | − t,ω ( z ) (cid:46) (cid:88) z ∈ N P ∩ ω ∗ h r + t ) z | r P | r,ω ( z ) (cid:46) ||| u P − u P (cid:48) ||| + (cid:107) h tP ( r P − w ) (cid:107) ω ∗ + (cid:107) h tP ( r P − v ) (cid:107) ω ∗ + (cid:88) z ∈ N P ∩ ω ∗ h r + t ) z | r P − v | r,ω ( z ) , (115) where h z = min { τ : z ∈ τ } h τ . Moreover, for any v ∈ S P , we have the global bounds (cid:88) z ∈ N P | r P | − t,ω ( z ) (cid:46) (cid:88) z ∈ N P h r + t ) z | r P | r,ω ( z ) (cid:46) ||| u − u P ||| + (cid:107) h tP ( r P − v ) (cid:107) + (cid:88) z ∈ N P h r + t ) z | r P − v | r,ω ( z ) . (116) Proof.
Set s = − t >
0. Let e = u P (cid:48) − u P , and let e P ∈ S P be the L -orthogonal projectionof e onto S P . Then for any v ∈ S P (cid:48) , we have (cid:104) Ae, e (cid:105) = (cid:104) r P , e (cid:105) = (cid:104) r P , e − e P (cid:105) = (cid:104) r P , e (cid:105) Γ ∗ = (cid:104) r P − v, e (cid:105) Γ ∗ + (cid:104) v, e (cid:105) Γ ∗ ≤ (cid:107) h − sP (cid:48) ( r P − v ) (cid:107) Γ ∗ (cid:107) h sP (cid:48) e (cid:107) Γ ∗ + (cid:107) v (cid:107) H s (cid:107) e (cid:107) ˜ H − s , (117)where we have used the fact that e − e P = 0 outside Γ ∗ . Upon using the inverse inequality (cid:107) h sP (cid:48) e (cid:107) (cid:46) (cid:107) e (cid:107) ˜ H − s , this gives ||| e ||| (cid:46) (cid:107) h − sP (cid:48) ( r P − v ) (cid:107) Γ ∗ + (cid:107) v (cid:107) H s (cid:46) (cid:107) h − sP (cid:48) ( r P − v ) (cid:107) Γ ∗ + (cid:107) φ r P − v (cid:107) H s + (cid:107) φ r P (cid:107) H s . (118)To localize the last term, we follow an approach from Carstensen et al. (2001). With φ z ∈ S P the standard nodal basis function at z ∈ N P , we have (cid:107) φ r P (cid:107) H s (cid:46) (cid:88) z ∈ N P ∩ ω ∗ (cid:107) φ z r P (cid:107) H s ( ω ( z )) . (119)The linear operator T : H s ( ω ( z )) → H s ( ω ( z )) defined by T f = ( f − (cid:104) f, (cid:105) ω ( z ) ) φ z satisfies (cid:107) T f (cid:107) H s (cid:46) | f | H s ( ω ( z )) , cf. Carstensen et al. (2001). Hence (cid:107) φ z r P (cid:107) H s = (cid:107) φ z r P − φ z (cid:104) r P , (cid:105) ω ( z ) (cid:107) H s (cid:46) (cid:107) r P (cid:107) H s ( ω ( z )) , (120)where we have taken into account that r P ⊥ S P , and (114) follows.For the lower bounds, we first prove a couple of inequalities involving the auxiliaryerror indicator (cid:107) h − sP r P (cid:107) τ . Let w ∈ S P (cid:48) , and let w P ∈ S P be the L -orthogonal projectionof h − sP w onto S P . Then we have (cid:104) h − sP w, h − sP w (cid:105) = (cid:104) w − r P , h − sP w (cid:105) + (cid:104) A ( u P (cid:48) − u P ) , h − sP w − w P (cid:105)≤ (cid:104) w − r P , h − sP w (cid:105) + (cid:107) A ( u P (cid:48) − u P ) (cid:107) H s (Γ ∗ ) (cid:107) h − sP w − w P (cid:107) ˜ H − s (Γ ∗ ) (cid:46) (cid:107) h − sP ( w − r P ) (cid:107) supp w (cid:107) h − sP w (cid:107) + (cid:107) u P (cid:48) − u P (cid:107) H − s (Γ) (cid:107) h − sP w (cid:107) Γ ∗ , (121) Negative order operators 27 where in the last step we used (22), and so (cid:107) h − sP r P (cid:107) supp w (cid:46) ||| u P − u P (cid:48) ||| + (cid:107) h − sP ( r P − w ) (cid:107) supp w . (122)Let v ∈ L , and let v P ∈ S P be the L -orthogonal projection of v onto S P . Using (22),then we have (cid:104) h − s r P , h s v (cid:105) = (cid:104) r P , v (cid:105) = (cid:104) A ( u − u P ) , v − v P (cid:105) (cid:46) (cid:107) A ( u − u P ) (cid:107) H s (Γ) (cid:107) v − v P (cid:107) ˜ H − s (Γ) (cid:46) ||| u − u P |||(cid:107) h s v (cid:107) , (123)establishing (cid:107) h − sP r P (cid:107) (cid:46) ||| u − u P ||| . (124)On the other hand, for any N ⊆ N P and v ∈ S P , we have (cid:88) z ∈ N h r − s ) z | r P | r,ω ( z ) (cid:46) (cid:88) z ∈ N (cid:16) h r − s ) z | v | r,ω ( z ) + h r − s ) z | r P − v | r,ω ( z ) (cid:17) (cid:46) (cid:88) z ∈ N (cid:16) (cid:107) h − sP v (cid:107) ω ( z ) + h r − s ) z | r P − v | r,ω ( z ) (cid:17) (cid:46) (cid:107) h − sP v (cid:107) γ + (cid:88) z ∈ N h r − s ) z | r P − v | r,ω ( z ) (cid:46) (cid:107) h − sP r P (cid:107) γ + (cid:107) h − sP ( v − r P ) (cid:107) γ + (cid:88) z ∈ N h r − s ) z | r P − v | r,ω ( z ) , (125)where γ = int (cid:83) z ∈ N ω ( z ). Then the second inequalities in (115) and (116) follow from (122)and (124), respectively. Finally, the first inequalities in (115) and (116) are a consequenceof the fact that r P is L -orthogonal to S P .Let us record some useful bounds on the various oscillation terms that appeared in thepreceding lemma. Lemma 5.2.
Let
P, P (cid:48) ∈ adm( P ) be with P (cid:22) P (cid:48) , and let γ = int (cid:83) τ ∈ Q τ with some Q ⊆ P . Assume that f ∈ H r (Γ) and that A : ˜ H r +2 t (Γ) → H r (Γ) is bounded for some r ∈ ( − t, ) ∩ (0 , − t ) ∩ [0 , ν ] , so that r P ∈ H r (Γ) . Then we have min w ∈ S P (cid:48) (cid:107) h tP ( r P − w ) (cid:107) γ (cid:46) (cid:88) τ ∈ Q h r + t ) τ | r P | r,τ . (126) With v = Q ˆ P r P ∈ S P the quasi-interpolant of r P , we also have (cid:107) h tP ( r P − v ) (cid:107) γ + (cid:88) z ∈ N P ∩ γ | r P − v | − t,ω ( z ) (cid:46) (cid:88) z ∈ N P ∩ γ h r + t ) z | r P | r,ω ( z ) , (127) Negative order operators 28 where h z = min { τ : z ∈ τ } h τ . Finally, with Γ ∗ and ω ∗ as in Lemma 5.1, there exists v ∈ S P (cid:48) such that v = 0 outside ω ∗ and that (cid:107) h tP (cid:48) ( r P − v ) (cid:107) ∗ + (cid:107) φ r P − v (cid:107) H − t (Γ) (cid:46) (cid:88) z ∈ N P ∩ ω ∗ h r + t ) z | r P | r,ω ( z ) . (128) Proof.
The estimate (126) is a standard direct estimate, and (127) follows from (cid:88) z ∈ N P ∩ γ (cid:107) r P − v (cid:107) − t,ω ( z ) (cid:46) (cid:88) z ∈ N P ∩ γ h r + t ) z | r P | r,ω ( ω (( z )) (cid:46) (cid:88) z ∈ N P ∩ γ h r + t ) z | r P | r,ω ( z ) , (129)where we have used (23), (19), and the local finiteness of the mesh.For (128), let v ∈ S P (cid:48) be defined by v = (cid:88) z ∈ N P (cid:48) ∩ ω ∗ (cid:104) r P , (cid:105) ω (cid:48) ( z ) φ z , (130)where φ z ∈ S P (cid:48) is the standard nodal basis function at z ∈ N P (cid:48) , and (cid:104)· , ·(cid:105) ω (cid:48) ( z ) is the L -innerproduct on the star ω (cid:48) ( z ) around z with respect to P (cid:48) . Then we have (cid:107) h tP (cid:48) ( r P − v ) (cid:107) ∗ (cid:46) (cid:88) z ∈ N P (cid:48) ∩ ω ∗ | r P | − t,ω (cid:48) ( z ) (cid:46) (cid:88) z ∈ N P ∩ ω ∗ | r P | − t,ω ( z ) , (131)and from r P ⊥ L S P we infer the bound (128) for its first term. For the second term, wehave φ r P − v = (cid:88) z ∈ N P (cid:48) ∩ ω ∗ (cid:0) r P − (cid:104) r P , (cid:105) ω (cid:48) ( z ) (cid:1) φ z , (132)and hence (cid:107) φ r P − v (cid:107) H − t (Γ) (cid:46) (cid:88) z ∈ N P (cid:48) ∩ ω ∗ (cid:13)(cid:13)(cid:0) r P − (cid:104) r P , (cid:105) ω (cid:48) ( z ) (cid:1) φ z (cid:13)(cid:13) H − t ( ω (cid:48) ( z )) . (133)As in the proof of Lemma 5.1, now by using the boundedness of f (cid:55)→ (cid:0) f − (cid:104) f, (cid:105) ω (cid:48) ( z ) (cid:1) φ z in H − t ( ω (cid:48) ( z )), and then by employing the orthogonality r P ⊥ L S P again, we establish theproof. Remark . Similarly to Remark 4.3, one can also derive someresults on the saturation assumption for the non-residual estimators from Erath, Ferraz-Leite, Funken, and Praetorius (2009a). We skip the details.Combining (114) and (128), for the Faermann indicators we get ||| u P − u P (cid:48) ||| (cid:46) (cid:88) z ∈ N P ∩ ω ∗ | r P | − t,ω ( z ) + (cid:88) z ∈ N P ∩ ω ∗ h r + t ) z | r P | r,ω ( z ) , (134) Negative order operators 29 and by (115), the entire right hand side of the preceding inequality is controlled by the lastterm alone. This leads to generalized weighted residual indicators, for which we alreadyhave the global bounds (116). Obviously, the case r = 1, treated in Feischl et al. (2011a,b)is computationally more attractive, but the remaining cases r ∈ ( − t,
1) become importantif for instance f ∈ H r (Γ) \ H (Γ).In the following, we fix some r ∈ ( − t, ) ∩ (0 , − t ) ∩ [0 , ν ] and assume that f ∈ H r (Γ),and that A : ˜ H r +2 t (Γ) → H r (Γ) is bounded. Then defining the error estimator η ( v, P, γ ) = (cid:88) z ∈ N P ∩ γ h r + t ) z | f − Av | r,ω ( z ) , (135)for γ ⊆ Γ and v ∈ S P , with η ( P, γ ) = η ( u P , P, γ ), in the context of Lemma 5.1 we have ||| u P − u P (cid:48) ||| ≤ β η ( P, Γ ∗ ) , (136)where β > z (cid:55)→ τ P ( z ) : N P → P for any P ∈ adm( P ), satisfying z ∈ τ P ( z ). For v ∈ S P , letosc r ( v, P ) = (cid:107) h tP ( f − Av − w ) (cid:107) + (cid:88) z ∈ N ˆ P h r + t ) z | f − Av − w | r, ˆ ω ( z ) , (137)where ˆ ω ( z ) is the star around z ∈ N ˆ P with respect to the partition ˆ P , and w = Q ˆ P ( f − Av ) ∈ S P is the quasi-interpolator of f − Av , defined by Q ˆ P g = (cid:88) z ∈ N ˆ P g z ( z ) φ z , (138) φ z ∈ S P is the standard nodal basis function at z , and g z ∈ P is the L -orthogonalprojection of g onto the affine functions on τ ˆ P ( z ). This quasi-interpolator, as a variationon the Cl´ement interpolator, is introduced in Oswald (1994). We remark that we willonly need the idempotence Q P = Q ˆ P , so for example the Scott-Zhang operator could havebeen employed instead. Let g (cid:48) = g − Q ˆ P g with g ∈ H r (Γ), and let y ∈ N P . Then byidempotence, we have | g (cid:48) | r,ω ( y ) = | g (cid:48) − Q ˆ P g (cid:48) | r,ω ( y ) (cid:46) (cid:88) z ∈ N ˆ P ∩ ω ( z ) | ( g (cid:48) − g (cid:48) z ( z )) φ z | r, ˆ ω ( z ) , (139)which implies by the boundedness of g (cid:48) (cid:55)→ ( g (cid:48) − g (cid:48) z ( z )) φ z in H r (ˆ ω ( z )), that (cid:88) y ∈ N P | g − Q ˆ P g | r,ω ( y ) (cid:46) (cid:88) z ∈ N ˆ P | g − Q ˆ P g | r, ˆ ω ( z ) . (140) Negative order operators 30
In particular, in the context of Lemma 5.1 we have ||| u − u P ||| (cid:46) α η ( P, Γ) ≤ ||| u − u P ||| + osc r ( u P , P ) , (141)where α > Q ˆ P there exists a constant C Q > (cid:107) h tP ( r P − Q ˆ P r P ) (cid:107) ω ( γ ) + (cid:88) z ∈ N ˆ P ∩ ˆ ω ( γ ) h r + t ) z | r P − Q ˆ P r P | r, ˆ ω ( z ) ≤ C Q η ( P, γ ) , (142)for any P ∈ adm( P ) and γ ⊆ Γ, with ˆ ω ( γ ) = (cid:83) { ˆ ω ( z ) : z ∈ N ˆ P ∩ γ } .In the rest of this section, we follow the pattern of the preceding two sections, and skipthe proofs that closely resemble those given in the previous sections. Lemma 5.4.
Assume that (cid:88) z ∈ N P h r + t ) z | Av | r,ω ( z ) ≤ C A ||| v ||| , v ∈ S P , (143) for P ∈ adm( P ) , with the constant C A = C A ( A, σ s , σ g ) . Then the followings hold.a) There is a constant C G > such that ||| u − u P ||| + osc r ( u P , P ) ≤ C G inf v ∈ S P (cid:0) ||| u − v ||| + osc r ( v, P ) (cid:1) , (144) for any P ∈ adm( P ) .b) There exists a constant λ > such that η ( P (cid:48) , Γ) ≤ (1 + δ ) η ( P, Γ) − λ (1 + δ ) η ( P, Γ ∗ ) + C δ ||| u P − u P (cid:48) ||| , (145) for any P, P (cid:48) ∈ adm( P ) with P (cid:22) P (cid:48) , and for any δ > , with C δ depending on δ ,where Γ ∗ = int (cid:83) τ ∈ P \ P (cid:48) τ .c) Let P, P (cid:48) ∈ adm( P ) be such that P (cid:22) P (cid:48) , and let Γ ∗ = int (cid:83) τ ∈ P \ P (cid:48) τ . If, for some µ ∈ (0 , ) it holds that ||| u − u P (cid:48) ||| + osc r ( u P (cid:48) , P (cid:48) ) ≤ µ (cid:0) ||| u − u P ||| + osc r ( u P , P ) (cid:1) , (146) then with θ = α (1 − µ ) C Q + β (1+2 C A C Q ) , we have η ( P, Γ ∗ ) ≥ θ η ( P, Γ) , (147) where α > and β > are the constant from (141) and (136) , respectively. Negative order operators 31
Remark . The estimate (143) is proved in Section 6 for a general class of singularintegral operators of negative order, under the hypothesis that ˆ A : H t + σ (Ω) → H − t + σ (Ω)is bounded for some σ > n , and that 0 < r + t < σ , where ˆ A is an extension of A to Ω ifΓ is open, and ˆ A = A if Γ = Ω. Recalling that Ω is a C ν − , -manifold, the condition on σ translates to ν > n − t , see Remark 6.2 for details. Therefore for n = 1 and n = 2 we needa C , curve or a surface, respectively, assuming that ≤ − t <
1. Note that if Γ is open,the boundary of Γ is allowed to be Lipschitz.
Proof of Lemma 5.4.
The proofs of a) and c) go along the same lines as the correspondingproofs from the previous section, cf. Lemma 4.4. In particular, the stability (142) and thelocality of Q ˆ P are important.Claim b) is established if we show that (cid:88) z ∈ N P (cid:48) ∩ Γ ∗ h r + t ) z | g | r,ω (cid:48) ( z ) ≤ λ (cid:88) y ∈ N P ∩ Γ ∗ h r + t ) y | g | r,ω ( y ) , g ∈ H r (Γ) , (148)with some constant λ <
1, where ω (cid:48) ( z ) is the star around z ∈ N P (cid:48) with respect to thepartition P (cid:48) . For τ, σ ∈ P (cid:48) , let I ( τ, σ ) denote the interaction term between τ and σ inthe Slobodeckij (double integral) norm of g . First of all, any diagonal term I ( τ, τ ) thatappears in the left hand side also appears in the right hand side, and the correspondingfactors satisfy h z ≤ λ h y with some constant λ <
1. Henceforth we concentrate on theoff-diagonal terms. Note that by symmetry the order of τ and σ is not important, andthat the number of occurrences of the particular (unordered) pair ( τ, σ ) in the left handside of (148) is equal to the number of z ∈ N P (cid:48) ∩ Γ ∗ satisfying τ, σ ⊂ ω (cid:48) ( z ). Suppose thatthe pair ( τ, σ ) appears in the left hand side exactly (cid:96) times, for (cid:96) ∈ [0 , n − τ and σ are contained in two triangles from P that share a k -face for some k ∈ [ (cid:96) − , n ], where k = n means that the two triangles coincide. If this face is in Γ ∗ , then the vertices of thisface give at least (cid:96) points y ∈ N P ∩ Γ ∗ such that τ, σ ⊂ ω ( y ), meaning that the same pairappears in the right hand side at least (cid:96) times. On the other hand, if the shared face is notin Γ ∗ , this would mean that τ and σ are triangles from P , and they interact through onlythe vertices on N P ∩ ∂ Γ ∗ . We also see that the corresponding factors h r + t ) z shrink since h z is defined by taking minimum as h z = min { τ ∈ P (cid:48) : z ∈ τ } h τ , proving the claim (148). Proposition 5.6.
Let the assumption (143) of Lemma 5.4 hold. Let
P, P (cid:48) ∈ adm( P ) beadmissible partitions with P (cid:22) P (cid:48) , and let Γ ∗ = int (cid:83) τ ∈ P \ P (cid:48) τ . Suppose, for some ϑ ∈ (0 , that η ( P, Γ ∗ ) ≥ ϑ η ( P, Γ) . (149) Then there exist constants γ ≥ and µ ∈ (0 , such that ||| u − u P (cid:48) ||| + γ η ( P (cid:48) , Γ) ≤ µ (cid:0) ||| u − u P ||| + γ η ( P, Γ) (cid:1) . (150) Negative order operators 32
Gearing towards a convergence rate analysis, for u ∈ ˜ H t (Γ) the solution of Au = f with f ∈ H r (Γ), and for P ∈ adm( P ), we definedist r ( u, S P ) = inf v ∈ S P (cid:0) ||| u − v ||| + osc r ( v, P ) (cid:1) . (151)Furthermore, for ε > r ( u, ε ) = min { P − P : P ∈ adm( P ) , dist r ( u, S P ) ≤ ε } . (152)We have the following result on the D¨orfler marking, whose proof is entirely analogous tothe proof of Proposition 4.7. Proposition 5.7.
Let the assumption (143) of Lemma 5.4 hold. Let P ∈ adm( P ) , andlet θ ∈ (0 , θ ∗ ) with θ ∗ = α β (1+2 C A ) . Suppose that R ⊆ P is a subset whose cardinality isminimal up to a constant factor, among all R ⊆ P satisfying η ( P, Γ ∗ ( R )) ≥ θ η ( P, Γ) . (153) where Γ ∗ ( R ) = int (cid:83) τ ∈ R τ . Then we have R (cid:46) card r ( u, ε ) , (154) where ε is defined by ε = θ ∗ − θ C G θ ∗ (cid:0) ||| u − u P ||| + osc r ( P, Γ) (cid:1) . (155)Now let us specify our adaptive algorithm. Algorithm 3:
Adaptive BEM ( t < parameters : conforming partition P , and θ ∈ [0 , output : P k ∈ adm( P ) and u k ∈ S P k for all k ∈ N for k = 0 , , . . . do Compute u k ∈ S P k as the Galerkin approximation of u from S P k ; Identify a minimal (up to a constant factor) set R k ⊂ P k of triangles satisfying η ( P k , Γ ∗ ) ≥ θ η ( P k , Γ) , (156)where r k = f − Au k and Γ ∗ = int (cid:83) τ ∈ R k τ ; Set P k +1 = refine( P k , R k ); endfor Finally, we introduce the approximation class A r,s ⊂ A − ( H r (Γ)) with s ≥ u for which | u | A r,s = sup ε> (card r ( u, ε ) s ε ) < ∞ , (157)and record that our adaptive BEM produces optimally converging approximations. Inverse-type inequalities 33
Theorem 5.8.
Let the assumption (143) of Lemma 5.4 hold, and in Algorithm 3, supposethat θ ∈ (0 , θ ∗ ) with θ ∗ = α β (1+2 C A ) . Let f ∈ H r (Γ) and u ∈ A r,s for some s > . Thenwe have ||| u − u k ||| + osc r ( P k , Γ) ≤ C | u | A r,s ( P k − P ) − s , (158) where C > is a constant. In this section we shall justify the inverse-type inequality (27), which has been used inLemmata 3.3, 4.4, and 5.4, and hence played a crucial role in our analysis. We allow ageneral class of singular integral operators, specified by the assumptions that follow.We keep the assumptions formulated in Section 2 still in force. We will be concernedonly with closed manifolds (i.e., Γ = Ω), since the case of open surfaces Γ ⊂ Ω would followby restriction. In addition, we assume that Ω is embedded in some Euclidean space R N , sothat the Euclidean distance function dist : Ω × Ω → [0 , ∞ ) is well defined. Instead of theoperator A that featured in the previous sections, we will consider in this section a moregeneral bounded linear operator T : H t (Ω) → H − t (Ω), hence removing the self-adjointnessand coerciveness assumptions. With ∆ = { ( x, x ) : x ∈ Ω } the diagonal of Ω × Ω, we assumethat there is a kernel K ∈ L (Ω × Ω \ ∆) associated to T , meaning that (cid:104) T u, v (cid:105) = (cid:104) K, u ⊗ v (cid:105) , (159)whenever u, v ∈ C ν − , (Ω) have disjoint supports. We assume that K is smooth onΣ = { τ × τ (cid:48) : τ, τ (cid:48) ∈ P } \ ∆ ⊂ Ω × Ω \ ∆ , (160)satisfying the estimate | ∂ αξ ∂ βη K ( ξ, η ) | ≤ C α,β dist( ξ, η ) n +2 t + | α | + | β | , ( ξ, η ) ∈ Σ , (161)for all multi-indices α and β satisfying n + 2 t + | α | + | β | >
0. Note that the partialderivatives are understood in local coordinates (or, as we discussed in §
2, in terms of thereference triangles). The kernels satisfying this smoothness condition have been called standard kernels , e.g., in Dahmen, Harbrecht, and Schneider (2006). Then one can showthat the kernels of a wide range of boundary integral operators are standard kernels, cf.Schneider (1998).
Theorem 6.1.
With T ∗ denoting the adjoint of T , let both T, T ∗ : H t + σ (Ω) → H − t + σ (Ω) be bounded for some σ > n . Moreover, assume s ≥ and < s + t < σ . Let S dP ⊂ H t (Ω) .Then we have (cid:88) τ ∈ P h s + t ) τ (cid:107) T v (cid:107) H s ( τ ) (cid:46) (cid:88) z ∈ N P h s + t ) z (cid:107) T v (cid:107) H s ( ω ( z )) (cid:46) (cid:107) v (cid:107) H t (Ω) , v ∈ S dP , (162) Inverse-type inequalities 34 for P ∈ adm( P ) , where N P is the set of all vertices in the triangulation P , ω ( z ) is thestar around z with respect to P , and h z = min { τ : z ∈ τ } h τ . As already mentioned, note that taking v = 0 on a subcollection of triangles in theabove theorem allows us to treat the case of open surfaces Γ ⊂ Ω. Remark . The boundedness of both
T, T ∗ : H t + σ (Ω) → H − t + σ (Ω) has been proved for σ < for general boundary integral operators on Lipschitz surfaces in Costabel (1988),and the endpoint case σ = is established for boundary integral operators associated tothe Laplace operator on Lipschitz domains in Verchota (1984). Unfortunately, we see thatthe preceding results require more than σ = . Since ν ≥ σ + | t | , we necessarily have ν > n + | t | . This allows Lipschitz curves for | t | < , and C , curves and surfaces for | t | <
1. Even though in general it rules out polyhedral surfaces, note that for the case of anopen surface Γ, its boundary can be Lipschitz polygonal, as long as one can find a smoothenough manifold Ω with Γ ⊂ Ω. As for the question of whether the boundedness holdsfor σ < ν − for C ν − , domains, let us note that the main ingredients of the results inCostabel (1988) are the near-optimal trace theorem for Lipschitz domains, which appearsin Costabel (1988) and Ding (1996), and a certain regularity result for the Poincar´e-Steklovoperator for Lipschitz domains, which appears, e.g., in McLean (2000). The relevant versionof the trace theorem has been proved for C ν − , domains in Kim (2007), see also Marschall(1987). For the regularity of the Poincar´e-Steklov operator, the author has not been ableto locate in the literature a result strong enough to give the boundedness for σ < ν − ,although there are results, e.g., in McLean (2000), that imply σ = ν − T v into the part that is in S dP , which we call the low frequency part, andits complement, which we call the high frequency part. The low frequency part is readilyhandled by either the standard inverse estimates or the new inverse estimates from Dahmen,Faermann, Graham, Hackbusch, and Sauter (2004). To treat the high frequency part, weintroduce a wavelet basis for the complement of S dP , and as naturally suggested by thetechniques we use, the high frequency part is further decomposed into terms correspondingto far-field , near-field , and local interactions. Please be warned that these names are onlysuggestive in that, e.g., the local interaction terms may contain interactions between twowavelets with non-overlapping supports, although they cannot be too far apart.Our main analytic tool is a locally supported wavelet basis for the energy space H t ,with the dual multiresolution analysis based on piecewise polynomial-type spaces. Morespecifically, we assume that there is a Riesz basis Ψ = { ψ λ } λ ∈∇ of H t of wavelet type ,whose dual, denoted by ˜Ψ = { ˜ ψ λ } λ ∈∇ , is locally supported piecewise polynomial wavelets,where ∇ is a countable index set. Now we expand on what we mean exactly by the variousadjectives such as “wavelet type” that characterize the bases Ψ and ˜Ψ.The collections Ψ ⊂ H t (Ω) and ˜Ψ ⊂ H − t (Ω) are biorthogonal: (cid:104) ψ λ , ˜ ψ µ (cid:105) = δ λµ , and are Inverse-type inequalities 35
Riesz bases for their corresponding spaces, meaning that (cid:13)(cid:13)(cid:80) λ ∈∇ v λ ψ λ (cid:13)(cid:13) H t (Ω) (cid:104) (cid:107) ( v λ ) λ (cid:107) (cid:96) ( ∇ ) , (cid:13)(cid:13)(cid:13)(cid:80) λ ∈∇ v λ ˜ ψ λ (cid:13)(cid:13)(cid:13) H − t (Ω) (cid:104) (cid:107) ( v λ ) λ (cid:107) (cid:96) ( ∇ ) , (163)for any sequence ( v λ ) λ ∈ (cid:96) ( ∇ ). Here the notation X (cid:104) Y means Y (cid:46) X (cid:46) Y . Eachwavelet ψ λ or ˜ ψ λ has a scale, which is encoded by the function | · | : ∇ → N . We say that ψ λ and ˜ ψ λ have the scale 2 −| λ | , which is justified by the locality propertiesdiam(supp ψ λ ) (cid:46) −| λ | , diam(supp ˜ ψ λ ) (cid:46) −| λ | . (164)We will also assume that the wavelet supports are locally finite , in the sense that { λ : | λ | = (cid:96), B ( x, − (cid:96) ) ∩ supp ψ λ (cid:54) = ∅ } (cid:46) , (165)and similarly for the dual wavelets, where the bounds do not depend on (cid:96) ∈ N and x ∈ Ω,and B ( x, ρ ) = { y ∈ Ω : dist( x, y ) < ρ } . An immediate consequence of this property is that { λ ∈ ∇ : | λ | = (cid:96) } (cid:46) n(cid:96) .We assume that the wavelets have the so-called cancellation property of order p ∈ N ,saying that there exists a constant η >
0, such that for any q ∈ [1 , ∞ ], for all continuous,piecewise smooth functions v on P and λ ∈ ∇ , |(cid:104) v, ψ λ (cid:105)| (cid:46) −| λ | ( n − nq + t + p ) max τ ∈ P | v | W p,q ( B (supp ψ λ , −| λ | η ) ∩ τ ) , (166)where for A ⊂ R N and ε > B ( A, ε ) := { y ∈ R N : dist( A, y ) < ε } .Furthermore, we assume that for all r ∈ [ − p, γ ), s < γ , necessarily with | s | , | r | ≤ ν , (cid:107) w (cid:107) H r (Ω) (cid:46) (cid:96) ( r − s ) (cid:107) w (cid:107) H s (Ω) , for w ∈ span { ψ λ : | λ | = (cid:96) } , (167)with γ = sup { s : Ψ ⊂ H s (Ω) } , and similarly for the dual wavelets, with γ and p replacedby ˜ γ and ˜ p , respectively.We assume that the norm equivalence (cid:13)(cid:13)(cid:13)(cid:80) λ ∈∇ v λ ˜ ψ λ (cid:13)(cid:13)(cid:13) H s (Ω) (cid:104) (cid:88) λ ∈∇ s + t ) | λ | | v λ | , (168)is valid for s ∈ ( − ˜ p, p ) ∩ ( − γ, ˜ γ ). As far as the following proof of Theorem 6.1 is concerned,we will use only the“greater than” part of the first norm equivalence in (163), and the “lessthan” part of (168) with s equal to the same parameter in the theorem. For this and otherreasons, in what follows we assume that the parameters p , ˜ p , γ , and ˜ γ are sufficiently large.Such a possibility is guaranteed by the constructions in Dahmen and Schneider (1999), seethe remark below. Note that p and ˜ p are, respectively, ˜ d and d as compared to, e.g., Stevenson (2004). Inverse-type inequalities 36
For λ ∈ ∇ , we define Ω λ = Ω ∩ B λ , with B λ an open ball with diam( B λ ) (cid:46) −| λ | ,containing both supp ψ λ and supp ˜ ψ λ . Thus Ω λ can be thought of as a common support of ψ λ and ˜ ψ λ . We then defineΛ = { λ : Ω λ ∩ τ (cid:54) = ∅ and diam(Ω λ ) ≥ δh τ for some τ ∈ P } , (169)where δ > λ ∩ τ (cid:54) = ∅ and λ (cid:54)∈ Λ imply Ω λ ⊂ ω ( τ ) for τ ∈ P .Roughly speaking, the index set Λ corresponds to the wavelets that are needed to resolvethe finite element space S dP . Note that the existence of such a δ > P . On the wavelet equivalent of S dP , we assume the inverse inequality (cid:107) v (cid:107) H s (Ω) (cid:46) (cid:107) h − sP v (cid:107) , v ∈ S Λ := span { ˜ ψ λ : λ ∈ Λ } , (170)for s ∈ [0 , ν ]. By a duality argument (test v against w (cid:104) h sP v ) this implies (cid:107) h sP v (cid:107) (cid:46) (cid:107) v (cid:107) H − s (Ω) , v ∈ S Λ , (171)for s ∈ [0 , ν ]. Remark . Concrete examples of wavelet bases satisfying all our assumptions are givenby the duals of the bases constructed in Dahmen and Schneider (1999). On triangulationsover a Lipschitz polyhedral surface, one can also use the construction in Stevenson (2003).The only reason for insisting on polynomial dual wavelets is that in the proof of Lemma6.4 below, we use the inverse estimate (171), which is a consequence of (170). The latterestimate (hence both) can be proven by adapting the techniques from Dahmen et al. (2004).Now that we have settled on our main tool, we can start with
Proof of Theorem 6.1 .Let v ∈ S dP be as in the theorem. Then first we estimate the part of T v that is in S Λ . Wedefine the projection operator Q Λ : H − t (Ω) → S Λ by Q Λ (cid:88) λ ∈∇ w λ ˜ ψ λ = (cid:88) λ ∈ Λ w λ ˜ ψ λ . (172)In what follows, we will abbreviate the Sobolev norms as (cid:107) · (cid:107) s,ω = (cid:107) · (cid:107) H s ( ω ) and (cid:107) · (cid:107) s = (cid:107) · (cid:107) H s (Ω) . Lemma 6.4 (Low frequency) . Let s ≥ , and suppose that either t > or s + t > . Thenwe have (cid:88) z ∈ N P h s + t ) z (cid:107) Q Λ T v (cid:107) s,ω ( z ) (cid:46) (cid:107) v (cid:107) t for v ∈ H t (Ω) . (173) Proof.
First let t ≤
0, and therefore s + t >
0. Then we have (cid:88) z ∈ N P h s + t ) z (cid:107) Q Λ T v (cid:107) s,ω ( z ) (cid:46) (cid:88) z ∈ N P (cid:107) Q Λ T v (cid:107) − t,ω ( z ) ≤ (cid:107) Q Λ T v (cid:107) − t (cid:46) (cid:107) T v (cid:107) − t (cid:46) (cid:107) v (cid:107) t , (174) Inverse-type inequalities 37 where we have used in succession a standard inverse estimate, the super-additivity of theSobolev norms (3), the stability of Q Λ in H − t , and the boundedness of T .For the case t >
0, a standard inverse estimate gives (cid:88) z ∈ N P h s + t ) z (cid:107) Q Λ T v (cid:107) s,ω ( z ) (cid:46) (cid:88) z ∈ N P h tz (cid:107) Q Λ T v (cid:107) ω ( z ) ≤ (cid:88) z ∈ N P (cid:107) h tP Q Λ T v (cid:107) ω ( z ) (cid:46) (cid:107) h tP Q Λ T v (cid:107) . (175)At this point we employ the inverse estimate (171), to get (cid:107) h tP Q Λ T v (cid:107) (cid:46) (cid:107) Q Λ T v (cid:107) − t (cid:46) (cid:107) T v (cid:107) − t (cid:46) (cid:107) v (cid:107) t , (176)concluding the proof.What remains now is to bound ( I − Q Λ ) T v , which consists of only high frequencywavelets compared to what is in Q Λ T v ∈ S Λ . To this end, for λ ∈ Λ c := ∇ \ Λ, let usdefine (cid:96) λ by 2 − (cid:96) λ = max { h τ : τ ∈ P, τ ∩ Ω λ (cid:54) = ∅ } , (177)so that in light of the norm equivalence (168), we can write (cid:88) z ∈ N P h s + t ) z (cid:107) ( I − Q Λ ) T v (cid:107) s,ω ( z ) (cid:46) (cid:88) z ∈ N P h s + t ) z (cid:88) { λ ∈ Λ c :Ω λ ∩ ω ( z ) (cid:54) = ∅ } | λ | ( s + t ) | ( T v ) λ | (cid:46) (cid:88) z ∈ N P (cid:88) { λ ∈ Λ c :Ω λ ∩ ω ( z ) (cid:54) = ∅ } − (cid:96) λ ( s + t ) | λ | ( s + t ) | ( T v ) λ | (cid:46) (cid:88) λ ∈ Λ c | λ |− (cid:96) λ )( s + t ) | ( T v ) λ | , (178)where ( T v ) λ = (cid:104) T v, ψ λ (cid:105) is the coordinate of T v with respect to ˜ ψ λ , and in the last step wehave taken into account the fact that each Ω λ intersects with only a uniformly boundednumber of stars ω ( z ). Hence our aim is to bound the last expression in (178) by (cid:107) ( v λ ) λ (cid:107) (cid:96) for v ∈ S dP , where v λ = (cid:104) v, ˜ ψ λ (cid:105) .Before dealing with nonlocality of T , let us focus on the local properties. To this end,with S P,z = { v ∈ S dP : v = 0 outside ω ( z ) } , define Q z to be the L -orthogonal projec-tor onto S P,z in case of piecewise constants, and otherwise to be the quasi-interpolationoperator onto S P,z , as discussed around (23). Here ω k ( z ) = ω ( ω k − ( z )) for k ≥ Lemma 6.5 (Local interactions) . Let s ≥ and let t + s < σ . Then we have (cid:88) z ∈ N P h s + t ) z (cid:107) ( I − Q Λ ) T Q z v (cid:107) s,ω ( z ) (cid:46) (cid:107) v (cid:107) t for v ∈ S dP . (179) Inverse-type inequalities 38
Proof.
Replacing v with Q z v in (178), we get (cid:88) z ∈ N P h s + t ) z (cid:107) ( I − Q Λ ) T Q z v (cid:107) s,ω ( z ) (cid:46) (cid:88) λ ∈ Λ c | λ |− (cid:96) λ )( s + t ) | ( T Q z v ) λ | . (180)In order to bound this, we view it as the (squared) (cid:96) (Λ c )-norm of a sequence, and wetest the sequence against a general sequence ( w λ ) λ ∈ (cid:96) (Λ c ). As a preparation to this, for w ∈ span { ψ λ : | λ | = (cid:96) } , we have |(cid:104) w, T Q z v (cid:105)| (cid:46) (cid:107) w (cid:107) t − σ (cid:107) T Q z v (cid:107) − t + σ (cid:46) − σ(cid:96) (cid:107) w (cid:107) t (cid:107) Q z v (cid:107) t + σ (cid:46) − σ(cid:96) +( t + σ ) (cid:96) z (cid:107) w (cid:107) t (cid:107) Q z v (cid:107) (cid:46) − σ(cid:96) +( t + σ ) (cid:96) z (cid:107) w (cid:107) t (cid:107) v (cid:107) ω ( z ) , (181)where (cid:96) z is defined by 2 − (cid:96) z = h z . Then assuming that t ≤
0, for w = (cid:80) λ w λ ψ λ with( w λ ) λ ∈ (cid:96) (Λ c ), we infer (cid:88) λ ∈ Λ c ( | λ |− (cid:96) λ )( s + t ) ( T Q z v ) λ w λ (cid:46) (cid:88) z ∈ N P (cid:88) (cid:96) ≥ (cid:96) z + c ( (cid:96) − (cid:96) z )( s + t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:42) (cid:88) {| λ | = (cid:96) :Ω λ ∩ ω ( z ) (cid:54) = ∅ } w λ ψ λ , T Q z v (cid:43)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:88) z ∈ N P (cid:88) (cid:96) ≥ (cid:96) z + c ( (cid:96) − (cid:96) z )( s + t − σ )+ (cid:96) z t (cid:88) {| λ | = (cid:96) :Ω λ ∩ ω ( z ) (cid:54) = ∅ } | w λ | (cid:107) v (cid:107) ω ( z ) (cid:46) (cid:88) z ∈ N P (cid:96) z t (cid:88) { λ ∈ Λ c :Ω λ ∩ ω ( z ) (cid:54) = ∅ } | w λ | (cid:107) v (cid:107) ω ( z ) (cid:46) (cid:107) w (cid:107) t (cid:107) h − tP v (cid:107) (cid:46) (cid:107) w (cid:107) t (cid:107) v (cid:107) t , (182)where c ∈ R is a constant that depends on δ . This establishes the lemma.The case t > |(cid:104) w, T Q z v (cid:105)| (cid:46) − σ(cid:96) + σ(cid:96) z (cid:107) w (cid:107) t (cid:107) v (cid:107) t,ω ( z ) , (183)instead of (181).To deal with nonlocality of T , we will employ the Schur test, which we recall here. Lemma 6.6 (Schur test) . Let Λ and M be countable sets, and let (cid:96) (Λ , w ) be the weighted (cid:96) -space with the norm (cid:107) v (cid:107) (cid:96) (Λ ,w ) = (cid:107){ v λ w λ } λ ∈ Λ (cid:107) (cid:96) (Λ) , (184) where { w λ } λ ∈ Λ is a given positive sequence of weights. For a matrix with entries T λµ , λ ∈ Λ , µ ∈ M , we consider its boundedness as a linear operator T : (cid:96) ( M ) → (cid:96) (Λ , w ) .Suppose that there exist two positive sequences { θ λ } λ ∈ Λ and { ω µ } µ ∈ M , and two numbers α, β ∈ R such that (cid:88) µ ∈ M | T λµ | ω µ ≤ α θ λ , (cid:88) λ ∈ Λ | T λµ | w λ θ λ ≤ β ω µ . (185) Inverse-type inequalities 39
Then we have (cid:107)
T v (cid:107) (cid:96) (Λ ,w ) ≤ αβ (cid:107) v (cid:107) (cid:96) ( M ) , v ∈ (cid:96) ( M ) . (186)With the Riesz bases Ψ ⊂ H t (Ω) and ˜Ψ ⊂ H − t (Ω) at hand, the operator T : H t (Ω) → H − t (Ω) can be thought of as the bi-infinite matrix T : (cid:96) ( ∇ ) → (cid:96) ( ∇ ) with the elements T λµ = (cid:104) ψ λ , T ψ µ (cid:105) . The following estimate on these elements, established in Stevenson(2004), will be crucial: | T λµ | = |(cid:104) ψ λ , T ψ µ (cid:105)| (cid:46) (cid:32) −|| λ |−| µ || / δ ( λ, µ ) (cid:33) n +2 t +2 p (187)where δ ( λ, µ ) = 2 min {| λ | , | µ |} dist(Ω λ , Ω µ ) . (188)See also Schneider (1998) and Dahmen and Stevenson (1999) for earlier derivations for lessgeneral cases.Let us fix a constant ε >
0, whose value is to be chosen later. The following resultshows that the far-field terms behave rather well. Note that the condition p + t > p at will. Lemma 6.7 (Far-field interactions) . Define the operator F : H t (Ω) → H − t (Ω) with matrixelements F λµ = T λµ for λ ∈ Λ c and µ ∈ ∇ satisfying dist(Ω λ , Ω µ ) ≥ ε max { − (cid:96) λ , −| µ | } ,and F λµ = 0 otherwise. Let s + t > and p + t > . Then we have (cid:88) λ ∈ Λ c | λ |− (cid:96) λ )( s + t ) | ( F v ) λ | (cid:46) (cid:107) v (cid:107) t for v ∈ H t (Ω) . (189) Proof.
We apply the Schur test (Lemma 6.6) with θ λ = 2 −| λ | n/ − ( | λ |− (cid:96) λ )( d + t ) , ω µ = 2 −| µ | n/ ,and w λ = 2 ( | λ |− (cid:96) λ )( s + t ) . First we shall bound (cid:88) µ ∈∇ | F λµ | −| µ | n/ (cid:46) (cid:88) m ∈ N − mn/ (cid:88) {| µ | = m, F λµ (cid:54) =0 } (cid:32) −|| λ |− m | / δ ( λ, µ ) (cid:33) n +2 t +2 p , (190)by a multiple of θ λ . In the inner sum, we must sum over all µ such that δ ( λ, µ ) (cid:38) | λ |− (cid:96) λ when | µ | = m ≥ | λ | , and such that δ ( λ, µ ) (cid:38) max { ,m − (cid:96) λ } when | µ | = m ≤ | λ | . By localityof the wavelets and the Lipschitz property Ω, for λ ∈ ∇ , m ∈ N and β >
0, we have (cid:88) { µ : | µ | = m, δ ( λ,µ ) ≥ R } δ ( λ, µ ) − ( n + β ) (cid:46) R − β n max { ,m −| λ |} , (191)which appears, e.g., in Stevenson (2004). This gives (cid:88) { µ : | µ | = m, δ ( λ,µ ) (cid:38) | λ |− (cid:96)λ } δ ( λ, µ ) − ( n +2 t +2 p ) (cid:46) − ( | λ |− (cid:96) λ )(2 p +2 t )+( m −| λ | ) n/ , (192) Inverse-type inequalities 40 for m ≥ | λ | , and (cid:88) { µ : | µ | = m, δ ( λ,µ ) (cid:38) max { ,m − (cid:96)λ } } δ ( λ, µ ) − ( n +2 t +2 p ) (cid:46) − ( | λ |− (cid:96) λ )( p + t )+( m −| λ | ) n/ , (193)for m ≤ | λ | . By using the preceding estimates, we conclude (cid:88) µ ∈∇ | F λµ | −| µ | n/ (cid:46) (cid:88) m ≤| λ | − ( | λ |− (cid:96) λ )( p + t ) − ( | λ |− m )( p + t ) −| λ | n/ + (cid:88) m> | λ | − ( | λ |− (cid:96) λ )(2 p +2 t ) − ( m −| λ | )( p + t ) −| λ | n/ (cid:46) −| λ | n/ − ( | λ |− (cid:96) λ )( p + t ) . (194)Now we shall bound (cid:88) λ ∈ Λ c w λ θ λ | F λµ | (cid:46) (cid:88) (cid:96) (cid:88) {| λ | = (cid:96), F λµ (cid:54) =0 } ( (cid:96) − (cid:96) λ )(2 s + t − p ) − (cid:96)n/ (cid:32) −| (cid:96) −| µ || / δ ( λ, µ ) (cid:33) n +2 t +2 p , (195)by a multiple of ω µ . By construction, for λ ∈ Λ c and µ ∈ ∇ with F λµ (cid:54) = 0, we have2 − min { (cid:96) λ , | µ |} (cid:46) dist(Ω λ , Ω µ ), implying that2 − (cid:96) λ (cid:46) dist(Ω λ , Ω µ ) , and dist(Ω λ , Ω µ ) (cid:38) − min {| λ | , | µ |} . (196)In particular, the second estimate tells us that δ ( λ, µ ) (cid:38) (cid:96) ≤ | µ | , we estimate the inner sum as (cid:88) | λ | = (cid:96) ( (cid:96) − (cid:96) λ )(2 s + t − p ) − (cid:96)n/ | F λµ | ≤ (cid:88) | λ | = (cid:96) (cid:96) − (cid:96) λ )( s + t ) − (cid:96)n/ | F λµ | (cid:46) (cid:88) {| λ | = (cid:96), δ ( λ,µ ) (cid:38) } δ ( λ, µ ) s + t ) − (cid:96)n/ (cid:32) − ( | µ |− (cid:96) ) / δ ( λ, µ ) (cid:33) n +2 t +2 p (cid:46) − (cid:96)n/ − ( | µ |− (cid:96) )( n/ t + p ) = 2 − ( | µ |− (cid:96) )( t + p ) −| µ | n/ , (197)where we have used (cid:96) ≥ (cid:96) λ and p + t > s + t > s + t − p ≤
0, for (cid:96) ≥ | µ | , we estimate (cid:88) | λ | = (cid:96) ( (cid:96) − (cid:96) λ )(2 s + t − p ) − (cid:96)n/ | F λµ | ≤ (cid:88) | λ | = (cid:96) − (cid:96)n/ | F λµ | (cid:46) − (cid:96)n/ − ( (cid:96) −| µ | )( n/ t + p ) ( (cid:96) −| µ | ) n = 2 − ( (cid:96) −| µ | )( p + t ) −| µ | n/ . (198) Inverse-type inequalities 41
Now if 2 s + t − p >
0, for (cid:96) ≥ | µ | , we have (cid:88) | λ | = (cid:96) ( (cid:96) − (cid:96) λ )(2 s + t − p ) − (cid:96)n/ | F λµ | (cid:46) (cid:88) | λ | = (cid:96) ( (cid:96) −| µ | )(2 s + t − p ) δ ( λ, µ ) s + t − p − (cid:96)n/ | F λµ | (cid:46) ( (cid:96) −| µ | )(2 s + t − p ) − (cid:96)n/ − ( (cid:96) −| µ | )( n/ t + p ) ( (cid:96) −| µ | ) n = 2 − ( (cid:96) −| µ | )(2 p − s ) −| µ | n/ . (199)From the geometric decay of the preceding estimates, we infer (cid:88) λ ∈ Λ c ( | λ |− (cid:96) λ )(2 s + t − p ) −| λ | n/ | F λµ | (cid:46) −| µ | n/ , (200)which completes the proof.For the remaining terms, i.e., for the near-field terms, we employ the simple estimate (cid:104) ψ λ , T ψ µ (cid:105) (cid:46) (cid:107) ψ λ (cid:107) t − σ (cid:107) T ψ µ (cid:107) − t + σ (cid:46) (cid:107) ψ λ (cid:107) t − σ (cid:107) ψ µ (cid:107) t + σ (cid:46) −| λ | σ + | µ | σ . (201)We will see that this estimate gives sub-optimal results, that in general require the manifoldΩ to be smoother than Lipschitz. There exist sharper estimates in the literature, cf.Stevenson (2004); Dahmen et al. (2006), that exploit the piecewise smooth nature of thewavelets. However, the author was not able to make use of them to get better results.The best attempts so far by using the estimates from Dahmen et al. (2006) resulted inlogarithmic divergences, and the estimates from Stevenson (2004) in general need Ω to besmoother than Lipschitz, thus do not seem to give improvements in this regard. Lemma 6.8 (Near-field interactions) . Define the operator N : H t (Ω) → H − t (Ω) withmatrix elements N λµ = T λµ for λ ∈ Λ c and µ ∈ ∇ satisfying dist(Ω λ , Ω µ ) ≤ −| µ | and ε | µ | ≤ (cid:96) λ , and N λµ = 0 otherwise. Let σ > n . Then we have (cid:88) λ ∈ Λ c | λ |− (cid:96) λ )( s + t ) | ( N v ) λ | (cid:46) (cid:107) v (cid:107) t for v ∈ S dP . (202) Proof.
We apply the Schur test (Lemma 6.6) with θ λ = 2 (cid:37)(cid:96) λ − σ | λ | , ω µ = 2 ( (cid:37) − σ ) | µ | , and w λ = 2 ( | λ |− (cid:96) λ )( s + t ) , where (cid:37) = 2 σ − n >
0. We have (cid:88) µ ∈∇ | N λµ | ( (cid:37) − σ ) | µ | (cid:46) (cid:88) m ≤ (cid:96) λ − σ | λ | + (cid:37)m (cid:46) (cid:37)(cid:96) λ − σ | λ | , (203) Conclusion and References 42 and (cid:88) λ ∈ Λ c w λ θ λ | N λµ | (cid:46) (cid:88) { τ ∈ P :dist( τ, Ω µ ) ≤ −| µ | } (cid:88) (cid:96) ≥ (cid:96) τ (cid:96) − (cid:96) τ )( s + t ) (cid:37)(cid:96) τ − σ(cid:96) − σ ( (cid:96) −| µ | ) n ( (cid:96) − (cid:96) τ ) (cid:46) (cid:88) { τ ∈ P :dist( τ, Ω µ ) ≤ −| µ | } (cid:37)(cid:96) τ − σ(cid:96) τ − σ ( (cid:96) τ −| µ | ) = (cid:88) { τ ∈ P :dist( τ, Ω µ ) ≤ −| µ | } σ | µ |− n(cid:96) τ (cid:46) (cid:88) { τ ∈ P :dist( τ, Ω µ ) ≤ −| µ | } σ | µ | vol( τ ) (cid:46) ( σ − n ) | µ | = 2 ( (cid:37) − σ ) | µ | , (204)which establishes the proof.We end this section by assembling the promised proof. Proof of Theorem 6.1.
Let v ∈ S dP be as in the theorem, and consider the decomposition T v = Q Λ T v + ( I − Q Λ ) T v. (205)The first term on the right hand side is the low frequency part, treated in Lemma 6.4. Asdiscussed in (178), the second term is bounded as (cid:88) z ∈ N P h s + t ) z (cid:107) ( I − Q Λ ) T v (cid:107) s,ω ( z ) (cid:46) (cid:88) λ ∈ Λ c | λ |− (cid:96) λ )( s + t ) | ( T v ) λ | . (206)Comparing the definitions of the near- and far-field interactions, we see that the combina-tion of Lemma 6.7 and Lemma 6.8 takes care of all the contributions, except those comingfrom the components v µ with dist(Ω λ , Ω µ ) ≤ ε − (cid:96) λ and 2 −| µ | ≤ ε − (cid:96) λ . But by choosing ε > ε of the hidden constants in Lemma 6.7 and Lemma 6.8, since the choiceof ε > v ∈ S dP for P ∈ adm( P ). In this paper, we proved geometric error reduction for three kinds of adaptive boundaryelement methods for positive, negative, as well as zero order operator equations, withoutrelying on a saturation-type assumption. In fact, several types of saturation assumptionsfollow from our work as a corollary. Moreover, bounds on the convergence rates are obtainedthat are in a certain sense optimal.We established several new global- and local discrete bounds for a number of residual-type error estimators for positive, negative, as well zero order boundary integral equations,
Conclusion and References 43 including the estimators from Carstensen et al. (2001), Carstensen et al. (2004), and Faer-mann (2000, 2002). Some of our bounds contain oscillation terms, that give useful estimateson how far the current mesh is from saturation. In order to handle the oscillation terms,which turned out to be not strtaightforward, we introduced an inverse-type inequality in-volving boundary integral operators and locally refined meshes. Our proof of the inequalityin general requires the underlying surface Γ to be C , or smoother, but for open surfaces itallows the boundary of Γ to be Lipschitz. So in general, polyhedral surfaces are ruled out,which is very likely an artifact of the proof, since in Feischl et al. (2011b), the inequalityis proven for a model negative order operator on polyhedral surfaces.The current work gives rise to its fair share of open problems, and re-emphasizes someexisting ones. The following is an attempt at identifying the most pressing of them. • to prove the inverse-type inequality in § t ≥
0. This isestablished for Symm’s integral operator (with t = − ) in Feischl et al. (2011b). • to characterize the approximation classes associated to the proposed adaptive BEMs.Some progress on this question has been made in Aurada et al. (2012a). • to generalize the proofs to higher order boundary element methods. • to extend the analysis to transmission problems, and adaptive FEM-BEM coupling. • convergence rate for adaptive BEMs based on non-residual type error estimators. • complexity analysis, i.e., the problem of quadrature and linear algebra solvers. Inparticular, one would like to know how accurate the residual should be computed inthe error estimators. Acknowledgements
I would like to thank Dirk Praetorius for carefully reading an earlier version of thismanuscript, and for making several important comments. I thank the anonymous ref-erees for their reviews and suggestions. I also thank Michael Renardy, Nilima Nigam, andElias Pipping over at mathoverflow.net for pointing out the reference Kim (2007), andDoyoon Kim for making his paper available to me. This work is supported by an NSERCDiscovery Grant and an FQRNT Nouveaux Chercheurs Grant.
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