Technische Universität Graz

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1 Technische Universität Graz Convergence of adaptive 3D BEM for weaky singuar integra equations based on isotropic mesh-refinement M. Karkuik, G. Of, D. Praetorius Berichte aus dem Institut für Numerische Mathematik Bericht 2012/7

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3 Technische Universität Graz Convergence of adaptive 3D BEM for weaky singuar integra equations based on isotropic mesh-refinement M. Karkuik, G. Of, D. Praetorius Berichte aus dem Institut für Numerische Mathematik Bericht 2012/7

4 Technische Universität Graz Institut für Numerische Mathematik Steyrergasse 30 A 8010 Graz WWW: c Ae Rechte vorbehaten. Nachdruck nur mit Genehmigung des Autors.

5 Convergence of adaptive 3D BEM for weaky singuar integra equations based on isotropic mesh-refinement M. Karkuik 1 G. Of 2 D. Praetorius 1 1 Institute for Anaysis and Scientific Computing, Vienna University of Technoogy, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria 2 Institute of Computationa Mathematics, graz University of Technoogy, Steyrergaße 30, A-8010 Graz, Austria Michae.Karkuik@tuwien.ac.at, of@tugraz.at, Dirk.Praetorius@tuwien.ac.at Abstract We consider the adaptive owest-order boundary eement method (ABEM) based on isotropic mesh-refinement for the weaky-singuar integra equation for the 3D Lapacian. The proposed scheme resoves both, possibe singuarities of the soution as we as of the given data. The impementation thus ony deas with discrete integra operators, i.e. matrices. We prove that the usua adaptive mesh-refining agorithm drives the corresponding error estimator to zero. Under an appropriate saturation assumption which is observed empiricay, the sequence of discrete soutions thus tends to the exact soution within the energy norm. 1 Introduction The (h h/2)-error estimation strategy is a we known technique to derive a-posteriori errorestimates fortheerror φ Φ inthenaturaenergy norm; see[hnw]inthecontext of ordinary differentia equations, and the overview artice of Bank [B] or the monograph [A] in the context of the finite eement method: Let X be a discrete subspace of the energy space H and et X be its uniform refinement. With the corresponding Gaerkin soutions Φ and Φ, the canonica (h h/2)-error estimator η := Φ Φ (1) 5

6 is a computabe quantity [DLY] which can be used to estimate φ Φ, where φ H denotes the exact soution. For finite eement methods (FEM), the energy norm, e.g. = Γ ( ) L2 (Ω) in case of the Poisson-Dirichet probem, provides oca information, which eements of the underying mesh shoud be refined to decrease the error effectivey. For boundary eement methods (BEM), the energy norm is (equivaent to) a fractiona order (and possiby negative) Soboev norm and does typicay not provide oca information directy. Besides the (h h/2) strategy, severa a posteriori error estimators for the efficient numerica treatment of integra equations via adaptive mesh-refining agorithms are avaiabe in the iterature, see [CF] and the references therein. For exampe, estimators of residua type are proposed and anayzed in [CS95, CS96, C96, C97, CMS, CMPS], where weighted Soboev norms of integer order are used to ocaize norms of fractiona order. In [F98, F00, F02], fractiona order Soboev norms are considered on overapping subsets of a mesh, e.g. node patches. Furthermore, a projection of the residua onto mutieve functions may be used, as proposed in [MS, MSW], or other integra equations can be appied to estimate the error as in [SSt, S00]. Error estimators that account for p or hp-versions are even found in [CFS, H02, HMS01, HMS02]. In [EFGP, FP], ocaized variants of η were introduced for certain weaky-singuar and hypersinguar integra equations. However, the mentioned ist of works is by no means exhaustive. Recenty [FOP], convergence of some (h h/2)-steered adaptive mesh-refinement has been proved for inear mode probems in the context of FEM and BEM. In [AFP+], the concept of estimator reduction has been introduced to anayze convergence of anisotropic mesh-refinement steered by (h h/2)-type or averaging-based error estimators for weakysinguar integra equations arising in 3D BEM. However, in [AFP+, FOP] it is assumed that the right-hand side of the integra equation is computed anayticay. This assumption is reaxed in [AFGKMP] where, for the mixed boundary vaue probem with Dirichet and Neumann conditions in 2D, the resoution of the given data becomes a part of the adaptive oop. In this ast work [AFGKMP], the Dirichet data is assumed to be in H 1, and is, by the Soboev imbedding theorem, continuous. The approximation can thus be carried out by a noda interpoant, and approximation estimates from [C97, EFGP] can be used to extract oca information on the approximation error. Moreover, for the Neumann data, [AFGKMP] assumes additiona reguarity, and the Neumann data is then discretized by the L 2 -projection onto piecewise constants. Whereas the discretization of the Neumann data transfers directy to 3D BEM, the discretization of the Dirichet data does not. The reason is that H 1 -functions on a 2D manifod (i.e. the boundary of a 3D domain) ack continuity and hence noda interpoation is not aowed. The aim of this present work is twofod: First, we extend the one-dimensiona oca approximation estimates for noda interpoation from [C97, EFGP] to certain quasiinterpoation operators in two dimensions in Theorem 3. This theorem extends the approximation properties of quasi-interpoation operators on adaptive meshes to positive fractiona order Soboev spaces. It thus enabes us to contro the error that is induced by the approximation of the given data using appropriate data osciation terms whick ook simiar to the 2D case of [AFGKMP]. Since we are interested in the oca mesh size of 6

7 adaptivey refined meshes, it is not feasibe to use the interpoation theorem in an easy way to obtain approximation estimates in fractiona order Soboev spaces, which are, in fact, interpoation spaces. Instead, the essentia ingredient is an upper bound for the interpoation norm of weighted H 1 -spaces. To contro the discretization error, we use the(h h/2)-type error estimators from[fp]. We show that the sum of data osciation and error estimator is a means to contro energy error and data approximation in an efficient and, under the so-caed saturation assumption, reiabe way. For a Gaerkin boundary eement method in 3D, we then propose an adaptive agorithm which is steered by the sum of data osciation and error estimator. The second aim of this work is to show that the discrete soutions that are generated by the proposed adaptive agorithm converge to the exact soution. This is done by foowing the concept of estimator reduction from [AFP+]. With nothing but notationa overhead, this aows to transfer the resuts of [AFGKMP] from 2D to 3D. Using resuts from [AFP], convergence of adaptive FEM-BEM couping driven by(h h/2) estimators can be derived. The remainder of this paper is organized as foows: Section 2 introduces the mode probems as we as the integra formuations thereof. Then, the main resuts of this paper are summarized. In Section 3, we coect the preiminaries on Soboev spaces, boundary discretization, and discrete spaces. Furthermore, we prove approximation properties of H 1 -stabe projections in fractiona-order Soboev spaces. In Section 4, we introduce the error estimators as we as the data osciation. The adaptive agorithm is stated in Section 5, where we aso reca the newest-vertex-bisection refinement. The main resut of this paper is found in Theorem 15, which states the estimator reduction property and hence convergence of the proposed adaptive BEM agorithm. Finay, numerica experiments in Section 6 concude the work. 2 Mode Probem and anaytica resuts The aim of this section is to introduce the mode probem, its integra formuations, and the Gaerkin formuations. Afterwards, we give an overview on the main resuts contained in this work. For a stated properties of the integra operators invoved, we refer to the iterature, e.g., the monographs [HW, ML, SaS] Continuous mode probem. The considered mode probem is the Lapace equation: For Ω R 3 an open set with Lipschitz-boundary, it reads u = 0 in Ω, u = g on Γ := Ω, (2) where the inhomogeneous Dirichet data g H 1/2 (Γ) are given. The probem of finding u in equation (2) is equivaenty stated as foows: Find φ H 1/2 (Γ) such that Vφ = (K +1/2)g on Γ. (3) 7

8 Here, V is the simpe-ayer potentia and K is the doube-ayer potentia which are formay defined by Vψ(x) = ψ(y)g(x y) dγ(y), Γ Kv(x) = Cv(y) n(y) G(x y)dγ(y), where G( ) denotes the fundamenta soution of the 3D Lapacian Γ G(z) = 1 4π 1 z for z R 3 \{0}. The soution φ of (3) is then the norma derivative of u, i.e., φ = n u. Conversey, the soution of (3) provides a soution u of (2) by means of the representation formua, see e.g., [HW, Chapter 1.1]. Let, denote the L 2 -scaar product which is extended to duaity between H 1/2 (Γ) and H 1/2 (Γ). Note that V : H 1/2 (Γ) H 1/2 (Γ) is an eiptic and symmetric isomorphism between H 1/2 (Γ) and its dua H 1/2 (Γ). It thus provides a scaar product defined by φ,ψ = Vφ,ψ. We denote by :=, 1/2 the induced energy norm which is an equivaent norm on H 1/2 (Γ). Therefore, the theorem of Riesz-Fischer (resp. the Lax- Migram emma) proves the existence and uniqueness of the soution φ H 1/2 (Γ) of the variationa form φ,ψ = (K +1/2)g,ψ for a ψ H 1/2 (Γ). (4) 2.2. Gaerkin Discretization. We consider the owest-order Gaerkin discretization of (4) with the discrete space of piecewise constant functions P 0 (T ). Here, T is a reguar partition of Γ into trianges. Again, the Riesz-Fischer theorem provides a unique soution of the Gaerkin formuation Φ,Ψ = (K +1/2)g,Ψ for a Ψ P 0 (T ). (5) The right-hand side in the preceding equation can in principe be computed by methods proposed in [CPS, SaS, S]. However, the right-hand side in (5) invoves the doube-ayer potentia operator K. To decoupe the singuarities of K and possibe singuarities of g and to enabe the direct use of fast methods ike hierarchica matrices or the fast mutipoe method, we additionay approximate g by some appropriate G. To that end, we use the L 2 -projection Π : L 2 (Γ) S 1 (T ) and define G := Π g S 1 (T ), wheres 1 (T )isthespaceofpiecewise affine, gobaycontinuousfunctionsont. Therefore, we restrict ourseves to the use conforming trianguations of Γ. Then, we repace g by G and denote by Φ the corresponding Gaerkin soution of Φ,Ψ = (K +1/2)G,Ψ for a Ψ P 0 (T ). (6) 8

9 We stress that probem (6) is equivaent to a inear system Vx = ( 1 M + K)g, where x 2 and g are the coefficient vectors of Φ and G, V and K are the Gaerkin matrices for the discrete integra operators V and K, and M is a mass-type matrix. Remark. A remarkabe advantage of BEM over FEM is its possibe high-order approximation of the soution u of (2) by means of the represenation formua. For smooth φ and owest-order BEM, one can prove O(h 3 ) convergence for the pointwise error within Ω. However, this order is reduced to O(h 2 ) when using noda interpoation to discretize g as in [AFGKMP]. Contrary, discretization of g by L 2 -projection preserves O(h 3 ), see e.g., [St, Chapter 12.1] A-posteriori error estimation. The discretization error φ Φ is measured using the(h h/2)-type error estimators from[fp], where four different error estimators are provided. Lemma 6 recas resuts from the atter work: Under the saturation assumption φ Φ C sat φ Φ with some uniform constant 0 < C sat < 1, (7) the proposed error estimators τ provide ower and upper bounds of the Gaerkin error, C 1 eff τ φ Φ C re τ, (8) where ony the upper bound hinges on (7), and C eff,c re > 0 depend additionay on the shape reguarity of T and on Γ. Here, τ denotes any of the proposed error estimators, and Φ is the Gaerkin soution of (5) with respect to the uniform refinement T of T. Throughout, the upper index denotes quantities wich are ony of theoretica interest, but are not computed numericay. In a second step, we incude the data approximation error, which stems from approximating the right-hand side g by G. Under additiona reguarity g H 1 (Γ), we introduce some numericay computabe data osciation term osc := h 1/2 Γ (g G ) L2 (Γ) which measures the oca error of the data approximation g G H (Γ). Then, the proposed error estimators τ 1/2 now computed with respect to approximated Dirichet data fufi equation (8) up to osciation terms, i.e. it hods that C 1 eff τ φ Φ +osc and φ Φ +osc C re τ. (9) Throughout the work, the symbo abbreviates up to a mutipicative constant which may ony depend on shape reguarity of the mesh T. Moreover, abbreviates the inequaities and. Remark. The saturation assumption (7) mainy states that the adaptive scheme has reached an asymptotic phase [FP, Section 5.2]. However, it may fai to hod in genera, as was shown in [DN]. However, in [DN] it was shown that (7) hods if the data is sufficienty resoved, i.e., in the context of FEM there hods φ Φ C sat φ Φ +osc, 9

10 so that sma data osciation osc impies the saturation assumption. This observation is another reason for the incusion of the data osciation into the adaptive agorithm. We stress that the saturation assumption is usuay observed in experiments. A reason for this might be that we have to ensure (7) ony for the sequence of meshes that is generated by the numerica agorithm Adaptive mesh-refining agorithm. In Section 5, we introduce an adaptive mesh-refining agorithm which is steered by the oca contributions of the error estimator γ = ( µ 2 +osc 2 ) 1/2. Here, µ is a ocaized variant of the (h h/2)-based error estimator from (1). Theorem 15 then guarantees that the adaptive agorithm eads to im γ = 0. According to (9), the saturation assumption (7) for the non-perturbed probem thus yieds convergence of the discrete Gaerkin soutions Φ to the exact soution φ. 3 Preiminairies 3.1. Soboev spaces on the boundary. We assume throughout that Ω R 3 is a poyhedra domain. The usua Soboev spaces are denoted by L 2 (Ω) and H 1 (Ω). Soboev spaces with noninteger order are defined by use of the Soboev-Sobodeckij seminorm. Soboev spaces on the boundary Γ := Ω are defined ikewise by using a parametrization of Γ as a two-dimensiona manifod, see [SaS]. Equivaenty, noninteger order spaces can be defined as interpoation spaces. We use the K-method of interpoation, see [T], to that end. We stress that the definition of a noninteger order Soboev space as interpoation space yieds the same set of functions, but an equivaent norm. However, norm equivaence constants depend on the boundary Γ. Furthermore, we wi use weighted Soboev spaces. For a weight function w L (Γ) with w > 0 amost everywhere, we denote by H 1 (Γ,w) the space of a functions u H 1 (Γ) equipped with the norm u 2 H 1 (Γ,w) := wu 2 L 2 (Γ) + w Γu 2 L 2 (Γ). Here, Γ denotesthesurfacegradient. Accordingtothepropertiesofw,ithodsH 1 (Γ,w) = H 1 (Γ), but with a different norm. Throughout, we abbreviate the notation and write, e.g., H s or L2 instead of H s (Γ) and L2 (Γ), respectivey Discrete spaces. A mesh T on the boundary Γ = Ω consists of fat trianges which are denoted by T. We assume that T is reguar, i.e., it contains no hanging nodes. 10

11 Associated to every eement T is an affine eement map F T : T T, where T is a reference eement. The voume area defines the oca mesh-width h L by h T := h (T) := T 1/2, whereas ρ (T) is the diameter of the argest ba that can be inscribed in T. A sequence of meshes (T ) N is caed quasi-uniform if there are goba discretization parameters h and ρ such that h (T) h and ρ (T) ρ for a T T. We ca a sequence of meshes ocay quasi-uniform, if the mesh-size h (T) as we as the parameter ρ (T) are comparabe on adjacent eements, i.e. h (T) h (T ) and ρ (T) ρ (T ) for a T,T T with T T. We define the so-caed shape-reguarity constant by σ(t ) := max T T h (T) ρ (T). For γ > 0, we ca a sequence of meshes γ-shape-reguar if σ(t ) is bounded uniformy by γ, i.e. sup N σ(t ) γ. In this context, we wi aso ca a singe mesh quasi-uniform, ocay quasi-uniform, or γ-shape-reguar. Finay, γ-shape-reguar meshes are often referred to as isotropic meshes. We define N to be the set of nodes of a mesh T. For p N 0, poynomia spaces on the reference eement are denoted by P p ( T) := span { x i y k : 0 i+k p }. Spaces of piecewise poynomias are denoted by P p (T ) := { u L (Γ) : u F T P p ( T) for a T T }, S p (T ) := P p (T ) C 0 (Γ) The L 2 -projection. The L 2 -projections π : L 2 P 0 (T ) and Π : L 2 S 1 (T ) are defined by Obviousy, Π is bounded in L 2, i.e., (π ψ,ψ ) L2 = (ψ,ψ ) L2 for a Ψ P 0 (T ), (Π u,u ) L2 = (u,u ) L2 for a U S 1 (T ). Π u L2 u L2. Since S 1 (T ) H 1, one can ask if Π is aso stabe in H 1, i.e., Π u H 1 C u H 1 for a u H 1. (10) SincetheL 2 -normandh 1 -normareequivaent ons 1 (T ), thestabiity estimate(10)ceary hods with a constant C > 0 which a-priori depends on T. Forquasi-uniformmeshes T, itispossibe toshow thatc isindependent oft, see[bx]. We can expoit any Cemént-type quasi-interpoation operator J, e.g. the origina operator from [C], to see Γ Π u L2 Γ (Π u J u) L2 + Γ J u L2 h 1 Π u J u L2 + Γ u L2, 11

12 where we use the stabiity properties of J and an inverse inequaity. Here, h is the goba discretization parameter. Since Π is a projection, we concude Π u J u L2 = Π (u J u) L2 u J u L2 h Γ u L2. Now, (10) foows, and the constant C does not depend on the sequence of meshes T, but ony on the quasi-uniformity constants. Since this work is concerned with adaptive meshes, it is of interest to obtain (10) with a constant C which does not depend on the actua step of the adaptive agorithm even for ocay quasi-uniform meshes. We then say that the L 2 -projections Π are uniformy stabe in H 1. Likewise, we say that the L 2 -projection is uniformy stabe in H β for 1 2 β 1. There are severa works to this question, see [BPS, C01, C02, S01]. The anaysis in this work does not ony assume X +1 X, but aso X +1 X, where the X are the discrete spaces generated by the agorithm and the X are the discrete spaces on the uniformy refined meshes. To ensure this, wi use the cassica Newest-Vertex-Bisection agorithm for oca mesh-refinement. None of the above mentioned works can be used in this case, but a resut of the recent work [KPP] states the H 1 -stabiity of Π. Throughout, we wi assume that we are deaing with a sequence of uniformy shapereguar meshes T which are obtained by successive refinement, i.e. P 0 (T ) P 0 (T +1 ) Loca inverse estimates in H 1/2 and H 1/2. We wi need an inverse estimate in H 1/2 which even hods for quasi-uniform K-meshes, see [GHS, Theorem 3.6]. Lemma 1. It hods that h 1/2 Ψ L2 C 1 Ψ H 1/2 for a Ψ P 0 (T ). (11) The constant C 1 > 0 depends soey on an upper bound of σ(t ) and on Γ. Moreover, we sha need an inverse estimate in H 1/2, see [AKP]. Lemma 2. It hods that h 1/2 Γ U L2 C 2 U H 1/2 for a U S 1 (T ). (12) The constant C 2 > 0 depends soey on Γ Loca approximation estimate in H 1/2. In this section, we prove an approximation estimate in H 1/2, where as in the prior Section 3.4 the emphasis is aid on the fact that the right-hand side invoves the oca mesh-size h L. Theorem 3. For each continuous projection P : H α S 1 (T ), it hods that (1 P )g H α C 3 min { } h 1 α Γ g L2, h 1 α Γ (1 P )g L2 for a g H 1. The constant C 3 > 0 depends soey on 0 α < 1, the γ-shape-reguarity of T, the operator norm of P : H α H α, and on the boundary Γ. 12 (13)

13 Before we deveop the proof of (13), we first state two possibe choices for P in the foowing two remarks. Remark. The Scott-Zhang projection P from [SZ] can be defined in a way such that it is stabe in both L 2 and H 1. By interpoation arguments, P then is aso stabe in H α for 0 α 1, and its operator norm does depend soey on σ(t ). Remark. Suppose that the mesh-refinement ensures that the L 2 -projection Π onto S 1 (T ) is uniformy H β stabe for some 0 < β < 1 and that the stabiity estimate depends ony on σ(t ), e.g. newest vertex bisection is used throughout, cf. Section 5.1. By interpoation arguments, P = Π then is aso stabe in H α for 0 α < β, and its operator norm does depend soey on σ(t ). For the proof of Theorem 3, we define the ocay averaged mesh-size function h S 1 (T ) by h (z) = max { h (T) : T T with z T } (14) for a nodes z N. According to uniform shape-reguarity, h S 1 (T ) then is ocay equivaent to the oca mesh-size h P 0 (T ). Lemma 4. It hods pointwise on Γ that C5 1 h h C 5 h as we as C 1 6 Γ h C 6. (15) The constants C 5,C 6 > 0 depend ony on the γ-shape-reguarity of T. The heart of the proof of the approximation estimate (13) is the foowing estimate for the interpoation norm of H 1 ( h ) and H 1. Lemma 5. For u H 1 and a 0 < θ < 1, it hods that C 1 4 u [H 1 ( h ),H 1 ] θ h 1 θ Γ u L2 + h θ u L2. (16) The constants C 4 > 0 depends soey on θ and the surface area Γ of Γ. Proof. The interpoation norm of u is given by We first note that ( K(t,u) 2 = u 2 [H 1 ( h ),H 1 ] θ = inf u=u h +u 1 u h H 1 ( h ) +t u 1 H 1 0 t 2θ K(t,u) 2dt t. ) 2 inf u=u h +u 1 u h 2 H 1 ( h ) +t2 u 1 2 H 1 =: K 2(t,u) 2, 13

14 where the infimum is taken over a u h,u 1 H 1 ( h ) = H 1. Therefore, we may consider K 2 instead of K. We proceed as for the interpoation of weighted L 2 -spaces [T, Section 23] and choose the decomposition u(x) = u h (x)+u 1 (x) := ψ(x)u(x)+ ( 1 ψ(x) ) u(x), where ψ(x) := t 2 h2 (x)+t 2. We then have K 2 (t,u) 2 = Γ ψu 2 h2 + Γ (ψu) 2 h2 +t 2 (1 ψ)u 2 +t 2 Γ ((1 ψ)u) 2 dγ ψu 2 h2 +t 2 (1 ψ)u 2 dγ+ Γ (ψu) 2 h2 +t 2 Γ ((1 ψ)u) 2 dγ. Γ Γ For the integrand of the second integra in (17), we compute Γ (ψu) 2 h2 +t 2 Γ ((1 ψ)u) 2 = u Γ ψ +ψ Γ u 2 h2 +t 2 u Γ (1 ψ)+(1 ψ) Γ u 2 = h 2 u Γ ψ 2 +t 2 u 2 Γ (1 ψ) 2 + h 2 ψ Γ u 2 +t 2 (1 ψ) Γ u 2 +2 h 2 uψ Γψ Γ u+2t 2 u(1 ψ) Γ (1 ψ) Γ u. The ast ine in the preceding equation adds up to zero. This is seen from 2 h 2 uψ Γψ Γ u+2t 2 u(1 ψ) Γ (1 ψ) Γ u = 2u( h 2 ψ t2 (1 ψ)) Γ ψ Γ u (17) and h 2 ψ t2 (1 ψ) = 0 by definition of ψ. Therefore, (17) becomes K 2 (t,u) 2 ψu 2 h2 +t 2 (1 ψ)u 2 dγ+ h2 u Γ ψ 2 +t 2 u 2 Γ (1 ψ) 2 dγ Γ Γ + h2 ψ Γ u 2 +t 2 (1 ψ) Γ u 2 dγ, whence Γ u 2 [H 1 ( h ),H 1 ] θ t 2θ Γ t 2θ t 2θ ψu 2 h2 +t 2 (1 ψ)u 2 dγ dt t Γ Γ h2 u Γ ψ 2 +t 2 u 2 Γ (1 ψ) 2 dγ dt t h2 ψ Γ u 2 +t 2 (1 ψ) Γ u 2 dγ dt t. Let us compute the three parts separatey. Together with the identity (18) ψu 2 h2 +t 2 (1 ψ)u 2 = u 2 (t 2 h2 )/( h 2 +t 2 ), 14

15 substitution t = s h, and Fubini s theorem, the first doube integra becomes 0 t 2θ Γ ψu 2 h2 +t 2 (1 ψ)u 2 dγ dt t = Γ u 2 ( = 0 0 t 2θ t 2 h2 h2 +t 2 dt t dγ s 2θ+1 s 2 +1 ds )( Γ u 2 h 2θ+2 dγ Exacty the same arguments appy for the third doube integra in (18) and show t 2θ h2 ψ Γ u 2 +t 2 (1 ψ) Γ u 2 dγ dt ( 0 Γ t = s 2θ+1 ) 0 s 2 +1 )( Γ ds Γ u 2 h 2θ+2 dγ. Let us now compute the second doube integra in (18). Using the identity Γ (1 ψ) = Γ ψ = 2t2 h Γ h ( h 2 +t2 ) 2, the substitution t = s h, and Fubini s theorem, we see 0 t 2θ Γ h2 u Γ ψ 2 +t 2 u 2 Γ (1 ψ) 2 dγ dt t = Γ ( = 4 0 u 2 0 ). t 2θ4t4 h2 Γ h 2 ( h 2 +t2 ) 3 dt t dγ s 2θ+3 (s 2 +1) 3ds )( Γ u 2 h 2θ Γ h 2 dγ Finay, reca that Γ h 1 from Lemma 4. Using the estimates for the three parts on the right-hand side of (18), we thus arrive at u [H 1 ( h ),H 1 ] θ h 1 θ Now, h 1 concudes the proof of (16). u L2 + h 1 θ Γ u L2 + h θ u L2. Proof of Theorem 3. Let J denote an arbitrary Cément-type quasi-interpoation operator which satisfies a oca first-order approximation property as we as oca stabiity in H 1 h β (1 J )g L2 h 1+β Γ g L2 (19) h β Γ(1 J )g L2 h β Γg L2, (20) for a g H 1 and a β R. A vaid choice is, e.g., the Scott-Zhang projection from [SZ]. For this choice, the constants in (19) (20) depend ony on the γ-shape-reguarity of T. For β = 0, the estimates (19) (20) and h h give (1 J )g L2 g H 1 ( h ) as we as (1 J )g H 1 g H 1 ). 15

16 for a g H 1 = H 1 ( h ). Using the interpoation theorem for the operator (1 J ) : [H 1 ( h ),H 1 ] α H α and Lemma 5, we see (1 J )g H α g [H 1 ( h ),H 1 ] α h 1 α Γ g L2 + h α g L2 for a g H 1. Moreover, the projection property P J g = J g and stabiity of P yied (1 P )g H α (1 J )g H α + P (1 J )g H α (1 J )g H α for a g H 1. Combining the ast two estimates and using the identity (1 P )(1 J ) = (1 P ) and (1 J )g H 1, we may bootstrap this resuts to see (1 P )g H α = (1 P )(1 J )g H α h 1 α Γ (1 J )g L2 + h α (1 J )g L2 for a g H 1. By use of h h and the estimates (19) (20), we thus arrive at (1 P )g H α h 1 α Γ (1 J )g L2 + h α (1 J )g L2 h 1 α Γ g L2. In addition, we now may bootstrapthis estimate via (1 P ) 2 = (1 P ) and(1 P )g H 1 to see (1 P )g H α = (1 P )(1 P )g H α h 1 α Γ (1 P )g L2. Atogether, the combination of the ast two estimates concudes the proof. 4 A-posteriori error estimation Reca that Φ is the Gaerkin soution (5) with respect to the uniform refinement T of T. Likewise, Φ is the Gaerkin soution (6) with respect to T, computed from the same right-hand side as Φ, i.e.: Φ,Ψ = (K +1/2)G,Ψ for a Ψ P 0 (T ), Φ, Ψ = (K +1/2)G, Ψ for a Ψ P 0 ( T ). (21) We reca the foowing two resuts from [FP, Proposition 1.1]: Lemma 6. With η := Φ Φ, it hods that Moreover, the estimate η φ Φ. (22) φ Φ C reη (23) is equivaent to the saturation assumption (7) with C sat = ( 1 C 2 re) 1/2. 16

17 Lemma 7. The foowing four a-posteriori error estimators η := Φ Φ η := (1 π ) Φ µ := h 1/2 ( Φ Φ ) L2 µ := h 1/2 (1 π ) Φ L2. satisfy the equivaence estimates (24) η η C 7 σ(t ) µ and µ µ C 8 η. (25) The constants C 7,C 8 > 0 depend soey on Γ. We are now in position to prove estimate (9). Theorem 8. Assume that the sequence of meshes (T ) N aows for a sequence of L 2 - projections Π which is uniformy H β -stabe for some β > 1/2. Define the data osciations by osc := h 1/2 Γ (1 Π )g L2. Let τ {η, η,µ, µ }. Then, there is a constant C 9 > 0 which depends soey on Γ and γ-shape-reguarity of T, such that we have efficiency Under the saturation assumption (7), we have reiabiity where C 10 > 0 depends soey on Γ, γ, and C sat. C 1 9 τ φ Φ +osc. (26) C 1 10 φ Φ τ +osc, (27) Proof. The proof foows aong the ines of [AFGKMP] but is now transferred to 3D. By Lemma 7, it suffices to consider τ = η. According to the best-approximation property of the Gaerkin scheme, we have φ Φ φ Φ as we as Φ Φ Φ Φ respectivey Φ Φ Φ Φ. Using the triange inequaity and Lemma 6, we obtain η = Φ Φ η + Φ Φ φ Φ + Φ Φ φ Φ + Φ Φ φ Φ + (K )(g G ) H 1/2 φ Φ + g G H 1/2, where the first estimate foows from stabiity of Gaerkin schemes. Finay, Theorem 3 appied for P = Π and α = 1/2 shows estimate (26). In the same manner, the triange inequaity and Lemma 6 together with the saturation assumption (7) show φ Φ φ Φ + Φ Φ η + Φ Φ η + Φ Φ + Φ Φ η + g G H 1/2. Another appication of Theorem 3 shows the desired resut. 17

18 Remark. Theorem 8 aso hods if Φ and Φ are computed with different right-hand sides, namey To see this, note that since Φ,Ψ = (K +1/2)G,Ψ for a Ψ P 0 (T ) Φ, Ψ = (K +1/2)Ĝ, Ψ for a Ψ P 0 ( T ) g Ĝ H 1/2 (Γ) g G H 1/2 (Γ) + G Ĝ H 1/2 (Γ) g G H 1/2 (Γ), G Ĝ H 1/2 (Γ) = Π (G g) H 1/2 (Γ) g G H 1/2 (Γ). 5 Adaptive Mesh-refining agorithm In this chapter, we introduce the adaptive agorithm. For the oca mesh-refinement, we use the newest-vertex bisection agorithm see e.g. [V, Chapter 4] as we as Figure 1. The properties of the resuting mesh-refinement are coected in Section 5.1. Section 5.2 deas with the a-priori convergence of computed discrete soutions as we as the a-priori convergence of the approximated data. In Section 5.3, we finay state the adaptive agorithm. The main resut of this work is stated in Theorem 15: It states that the overa error estimator γ is contractive up to the norm of the difference of two successive Gaerkin soutions and the difference of two successive data approximations. Together with the a-priori convergence resuts from Section 5.2, we obtain that the adaptive agorithm drives the overa error estimator to zero. Under the saturation assumption (7), Theorem 8 finay yieds convergence of the computed discrete soutions towards the exact soution Newest Vertex Bisection. The newest vertex bisection agorithm is an edge-based refinement agorithm. For a given initia mesh T 0, one choses for every eement T T 0 a so-caed reference edge. Given a mesh T with a subset E of its edges, one performs the foowing steps to obtain T +1 : Agorithm 9 (NVB). Input: mesh T, set of marked edges E (0) := E, counter i := 0. Output: refined mesh T +1 (i) Define U (i) := T,T T T T E (i) (ii) If U (i), define E (i+1) { e E \E (i) e reference edge of T or T } := E (i) U (i), increase counter i i+1 and goto (i). 18

19 Figure 1: For each triange T T, there is one fixed reference edge, indicated by the doube ine (eft, top). Refinement of T is done by bisecting the reference edge, where its midpoint becomes a new node. The reference edges of the son trianges T T +1 are opposite to this newest vertex (eft, bottom). To avoid hanging nodes, one proceeds as foows: We assume that certain edges of T, but at east the reference edge, are marked for refinement (top). Using iterated newest vertex bisection, the eement is then spit into 2, 3, or 4 son trianges (bottom). If a eements are refined by three bisections (right, bottom), we obtain the so-caed uniform bisec(3)-refinement which is denoted by T. (iii) With E (i) being the marked edges, refine each eement T T according to the rues shown in Figure 1. In the next emma, we coect some properties of the newest-vertex bisection agorithm. Lemma 10. Consider a coarse mesh T 0 and a sequence of meshes (T ) N generated by agorithm NVB. Let T denote the uniform bisec(3)-refinement of T. Then, there hods the foowing: (i) is obtained by successive refinement, i.e., P 0 (T ) P 0 (T +1 ), (ii) the fine meshes are obtained by successive refinement, i.e., P 0 ( T ) P 0 ( T +1 ), (iii) is uniformy shape reguar, (iv) each eement T T \T +1 which is refined, is the union of its sons T T +1, i.e. T = { T T +1 : T T }. Moreover, there is a constant 0 < q < 1 with h +1 (T ) qh (T) for a sons T T +1 of a refined eement T T \T +1. (v) aows for uniformy H 1 -stabe L 2 -projections onto S 1 (T ). Proof. It is we known that the NVB agorithm fufis assumptions (i)-(iv). In the recent work [KPP] it is shown that the newest-vertex bisection agorithm does even aow for uniformy H 1 -stabe L 2 -projections onto S 1 (T ) A-priori convergence of Gaerkin soutions. The foowing eementary resut has aready been proved in [BV, Lemma 6.1]. It actuay states that the orthogona projections on any sequence of nested subspaces of some Hibert space a-priori converge strongy in the norm of the underying space. 19

20 Lemma 11. Let H be a Hibert space and X X +1 be a sequence of nested cosed subspaces of H. Let P : H X be the orthogona projection onto X and x H. Then, the imit x := im P x H exists and beongs to the cosure of X := =0 X with respect to H. Before we prove in Proposition 13 beow that our adaptive agorithm for the owestorder Gaerkin BEM (5) (6) eads to a-priori convergent sequences of Φ,Φ P 0 (T ), we first prove the a-priori convergence of the approximated data G = Π g. Lemma 12. Suppose that the sequence of meshes T satisfies Assumptions (i) and (v) of Lemma 10. Then, for given g H β and G := Π g, the L 2 -imit g := im G exists and satisfies g H β. Moreover, there hods weak convergence in H β as we as strong convergence for any 0 α < β. G g as (28) im G g H α = 0 (29) Proof. According to Lemma 11, the imit g := im G exists in L 2. Fix 0 < α < β. According to uniform H β -stabiity, the sequence (G ) is a bounded sequence in H β. Therefore, there is a weaky convergent subsequence (G k ) of (G ) such that G k g H β as k with a certain imit g H β. The Reich theorem now proves that the subsequence (G k ) of (G ) satisfies even strong convergence G k g H α. In particuar, this provides convergence in L 2, and the uniqueness of imits yieds equaity g = g. First, we thus obtain that g H β. Second, we can appy the same argument to see that each subsequence (G k ) of (G ) has a subsequence (G kj ) which converges weaky to g in H β. We argue by contradiction to see that this impies weak convergence (28) of the entire sequence: Assume that (G ) does not converge weaky to g. By definition, there is some functiona ψ H β, some scaar ε > 0, and some subsequence (G k ) of (G ) such that ψ(g k ) ψ(g ) ε Consequenty, each subsequence (G kj ) of (G k ) satisfies this estimate as we and thus cannot converge weaky to g. This, however, yieds a contradiction and proves G g weaky in H β. Finay, the Reich compactness theorem even predicts G g strongy in H α for α < β. 20

21 With the aid of the ast two emmata, we can show that any adaptive agorithm for the soution of (5) or (6) converges a-priori. Proposition 13. Suppose that the sequence of meshes T satisfies Assumptions (i) and (v) of Lemma 10. Let Φ and Φ be the Gaerkin soutions of (5) and (6). Then, there exist imits φ,φ H 1/2 such that im 0 Φ φ = 0 = im 0 Φ φ. The same hods under Assumptions (ii) and (v) of Lemma 10 for the fine mesh soutions Φ and Φ, and the imits are denoted by φ and φ. However, none of the imits φ,φ, φ, and φ H 1/2 coincide in genera. Proof of a-priori convergence of Φ resp. Φ. Reca that the simpe-ayer potentia V is inear, eiptic, continuous, and symmetric. Therefore, the Gaerkin projection which maps φ to Φ resp. Φ is the orthogona projection with respect to the energy scaar product,. Consequenty, Lemma 11 appies and provides imits φ and φ. Proof of a-priori convergence of Φ and Φ. Since Φ and Φ are computed with respect to the -dependent right hand side G, we note that Φ and Φ are not the orthogona projections of φ. Hence, we must not use Lemma 11 directy. Reca that the approximated data converge to some imit g by Lemma 12. Denote by Φ, the Gaerkin soution to (5) with right-hand side g. We may use Lemma 11 to see that the imit Φ, φ exists in H 1/2. Moreover, there hods φ Φ φ Φ, + Φ, Φ. Now, the first term tends to zero by definition, and the second term is bounded by stabiity of Gaerkin schemes Φ, Φ (1/2+K)(g G ) H 1/2 g G H 1/2 and thus tends to zero by Lemma 12. This proves a-priori convergence of Φ with imit φ in H 1/2. The same argument appies to the sequence of Gaerkin soutions Φ Convergent adaptive agorithm. The adaptive agorithm which we introduce beow, is steered by the oca refinement indicators γ (T) 2 := h 1/2 (1 π ) Φ 2 L 2 (T) + h1/2 Γ (g G ) 2 L 2 (T) = µ (T) 2 +osc (T) 2. (30) We stress that by use of the above choice for γ 2 = T T γ (T) 2, (31) thecomputationofthecoarse-meshsoutionφ isavoided, andony Φ hastobecomputed. The adaptive agorithm reads as foows: 21

22 Agorithm 14. Input: Initia mesh T 0, parameter θ (0,1), counter := 0. (i) Obtain T by uniform bisec(3)-refinement of T, see Figure 1. (ii) Compute soution Φ of (6) with respect to T. (iii) Compute refinement indicators γ (T) for a T T. (iv) Choose a set M T with minima cardinaity such that γ (T) 2 θ γ (T) 2 (32) T M T T with some fixed parameter 0 < θ < 1. (v) Set E to be the set of reference edges of the eements in M. (vi) Refine mesh T according to Agorithm 9 and obtain T +1. (vii) Update counter := +1 and goto (i). Now, we state the main resut of this section. Theorem 15. Agorithm 14 guarantees the existence of constants 0 < κ < 1 and C 11 > 0 such that γ +1 κγ +C 11 ( Φ+1 Φ 2 H 1/2 + Π +1 g Π g 2 H 1/2 ) 1/2 (33) for a N 0. In particuar, this impies estimator convergence imγ = 0. (34) 0 The constant κ = 1 (1 q)θ depends on the adaptivity parameter θ and the constant 0 < q < 1 from the mesh-refinement of Lemma 10. The constant C 11 > 0 depends soey on the initia mesh T 0 and on Γ. Proof. We consider γ +1. The triange inequaity yieds γ+1 2 = h 1/2 +1 (1 π +1) Φ +1 2 L 2 + h 1/2 +1 Γ(g Π +1 g) 2 L 2 ( h 1/2 +1 (1 π +1) Φ L2 + h 1/2 +1 (1 π +1)( Φ +1 Φ ) 2 ) L2 + h 1/2 +1 Γ(g Π +1 g) 2 L 2 ( h 1/2 +1 (1 π +1) Φ L2 + h 1/2 +1 ( Φ +1 Φ ) 2 1/2 ) L2 + h +1 Γ(g Π +1 g) 2 L 2, (35) where we have used the eementwise estimate (1 π +1 )( Φ +1 Φ ) L2 (T) Φ +1 Φ L2 (T) for a T T +1 22

23 in the second step. We now use Young s inequaity and the inverse inequaity of Lemma 1 to see, for arbitrary δ > 0, γ 2 +1 (1+δ) h 1/2 +1 (1 π +1) Φ 2 L 2 +C 2 1(1+δ 1 ) Φ +1 Φ 2 H 1/2 + h 1/2 +1 Γ(g Π +1 g) 2 L 2. By use of the inverse inequaity of Lemma 2, we proceed anaogousy for the osciation term, h 1/2 +1 Γ(g Π +1 g) 2 L 2 ( h 1/2 +1 Γ(g Π g) L2 + h 1/2 +1 Γ(Π +1 g Π g) L2 ) 2 so that utimatey (1+δ) h 1/2 +1 Γ(g Π g) 2 L 2 +C 2 2(1+δ 1 ) Π +1 g Π g 2 H 1/2, [ ] γ+1 2 (1+δ) h 1/2 +1 (1 π +1) Φ 2 L 2 + h 1/2 +1 Γ(g Π g) 2 L 2 +C11 2 (1+δ 1 ) [ Φ +1 Φ ] 2H + Π 1/2 +1 g Π g 2H 1/2 with C 11 = max{c 1,C 2 }. Now, we consider the first term of (35) T -eementwise, (T) := h 1/2 +1 (1 π +1) Φ 2 L 2 (T) + h1/2 +1 Γ(g Π g) 2 L 2 (T) for T T. For T M, there hods h +1 T qh T and (1 π +1 Φ ) = 0, and so (M ) q Γ (g Π g) 2 L 2 (T) qγ (M ) 2. T M h 1/2 Contrary, for T T \M it hods that (T) h 1/2 (1 π ) Φ 2 L 2 (T) + h1/2 Γ (g Π g) 2 L 2 (T) = γ (T) 2, ash h +1 andπ +1 beingabetterapproximationasπ. Hence, (T \M ) γ (T \M ) 2. Spitting the first term into marked and non-marked eements, we have (T ) = (M )+ (T \M ) qγ (M ) 2 +γ (T \M ) 2 = γ 2 +(q 1)γ (M ) 2 from which we concude, for a δ > 0, γ 2 +1 (1+δ) ( 1+(q 1)θ ) γ 2 +C 2 11(1+δ 1 ) ( 1+(q 1)θ ) γ 2, [ Φ +1 Φ 2H 1/2 + Π +1 g Π g 2H 1/2 ]. We now choose κ := ( 1+(q 1)θ ) < 1. Optimizing the choice of the free parameter δ > 0, we see that the previous estimate is equivaent to γ +1 κγ +C 11 ( Φ+1 Φ 2 H 1/2 + Π +1 g Π g 2 H 1/2 ) 1/2. 23

24 Since P 0 ( T ) P 0 ( T +1 ) by Lemma 10, we can appy Proposition 13 to ( Φ ) N and obtain Φ +1 Φ 2 H 1/2 0 as. Lemma 12 reveas Π +1 g Π g 2 H 1/2 0. We thus infer the perturbed contraction γ +1 κγ +o(1). This is the estimator reduction from[afp+, Lemma 2.3]. Again, it foows from eementary cacuus that γ 0 as. 6 Numerica Experiments 6.1. Specification of the probem. We consider the three-dimensiona L-shaped domain depicted in Fig. 2. It is composed of three cubes with side ength 0.5 and a common edge in the z-axis. The uniform initia mesh T 0 consists of 56 rectanguar trianges. We choose Figure 2: Initia mesh of the L shaped domain. the constant extension (in z-direction) of the singuar soution of the two-dimensiona L- shaped domain as soution of the considered three-dimensiona Dirichet boundary vaue probem (2): u(r,ϕ,z) = r 2/3 sin(2/3ϕ) (36) in cyindrica coordinates x = rcosϕ, y = rsinϕ, z = z. Thus, we have a non-smooth conorma derivative φ aong the reentrant edge, but the trace g of u is zero on the adjacent faces Computation of Gaerkin soution and preconditioning. The computations were performed by an impementation [OSW] of the Gaerkin boundary eement method based on semi-anaytic integration formuae[rs, Appendix C.2], the fast mutipoe method [GR], and an artificia mutieve preconditioner [S03] for the Gaerkin matrix of the simpeayer potentia. The atter is a modification of the BPX preconditioner [BPX, FS]. In case 24

25 of a sequence of uniformy refined meshes (T ) N0, the preconditioner for the mesh T L is based on the weighted sum L A s = h 2s (π π 1 ) =0 of L 2 projections π : L 2 P 0 (T ) (and π 1 = 0). Due to the spectra equivaence inequaities [O98, Theorem 2] c 1 w 2 H 1/2 A 1/2 w,w Γ c 2 L 2 w 2 H 1/2 for a w P 0 (T L ), (37) the operator A 1/2 is a suitabe preconditioner for the simpe-ayer potentia operator. The inversion of the Gaerkin matrix of A 1/2 can be avoided due to the identity ( A 1/2 h ) 1 = M 1 h A1/2 h M 1 h which invoves the inversion of the diagona mass matrices M h ony, where A 1/2 h [i,j] = A1/2 Ψ j,ψ i and M h [i,j] = Ψ k,ψ. This preconditioner has been extended to adaptivey refined meshes [O06, p. 69] by an extension to a uniform mesh. In practice, the artificia mutieve preconditioner does not utiize a sequence of nested meshed constructed by uniform refinement, but a sequence of artificia spaces constructed by the geometrica custering of the finest mesh as used for the fast mutipoe method Computation of upper error bound. Reiabiity of the proposed error estimators is (for the non-perturbed probem) equivaent to the saturation assumption (7), see (27). However, since we prescribe the exact soution φ L 2 (Γ), we can compute a reiabe error bound err. To that end, remember first that π : L 2 (Γ) P 0 (T ) is the L 2 (Γ)- orthogona projection. Using the triange inequaity and the best approximation property of the Gaerkin soution Φ with respect to the energy norm yieds φ Φ φ Φ + Φ Φ φ π φ + Φ Φ. Now, the second term is bounded by osc as in the proof of Theorem 8. To bound the first term, we use the approximation estimate [CP06, Theorem 4.1] for π and see φ π φ φ π φ H 1/2 (Γ) h 1/2 (φ π φ) L2 (Γ) h 1/2 (φ Φ ) L2 (Γ), where we used the T -piecewise best approximation property of π in the ast step. Setting err := h 1/2 (φ Φ ) L2 (Γ), we see φ Φ err +osc, (38) and hence the right-hand side provides reiabe feedback on the energy error. 25

26 6.4. Uniform refinement. In the case of uniform refinement, the sequence of meshes (T ) N0 is obtained by uniform bisec(3)-refinement, i.e. T +1 is a refinement of T, where a edges are bisected. The Φ P 0 (T ) are the soutions of the Gaerkin formuations (6) with right-hand side G S 1 (T ), whereas Φ are the soutions of the Gaerkin formuations (6) on the uniformy refined mesh T = T +1 with the same right-hand side, see (21). We define err unif, := err = h 1/2 (φ Φ ) L2 (Γ) osc unif, := osc = h 1/2 Γ (g G ) L2 (Γ) µ unif, := µ = h 1/2 (1 π ) Φ ) L2 (Γ). Carefuy note that the computation of Φ is, in principe, avoided by Agorithm 14. In fact, we compute it ony to obtain the reiabe error bound err unif, Adaptive refinement. In the case of adaptive refinement, the sequence of meshes (T ) N0 isobtainedbyempoyingagorithm14. Infact, thereisnoneedtostorethemeshes T, since the computation of Φ is avoided. We merey need to store T, consequenty we approximate the right-hand side g on these finer meshes. Atogether, the Gaerkin soution Φ P 0 ( T ) is obtained by soving We define Φ, Ψ = (K +1/2)Ĝ, Ψ for a Ψ P 0 ( T ). (39) err adap, := err = ĥ1/2 (φ Φ ) L2 (Γ) osc adap, := osc = ĥ1/2 Γ (g Ĝ) L2 (Γ) µ adap, := µ = h 1/2 (1 π ) Φ ) L2 (Γ). The same computation that resuted in (38) yieds the reiabe error bound φ Φ err adap, +osc adap, Comparison of uniform and adaptive approach. First of a, we compare the rate of convergence for uniform and adaptive approach by potting the invoved quantities over the number of degrees of freedom. For uniform refinement, we pot err unif,, µ unif,, and osc unif, over the number of boundary eements #T, where (T ) N0 is the sequence of uniform meshes. For adaptive refinement, we pot err adap,, µ adap,, and osc adap, over the number of boundary eements # T, where ( T ) N0 is the sequence of temporary, bisec(3)- refinedmeshes T ofthemeshes T generatedbyagorithm14. Notethat, incaseofuniform mesh refinement, the optima order of convergence of owest-order Gaerkin boundary eement methods is O(h 3/2 ) O(#T 3/4 ). However, the exampe is chosen in such a way that uniform mesh refinement can be predicted to exhibit a reduced order of convergence. 26

27 Furthermore, we pot the reiabe error bounds err unif, and err adap, over the time that is consumed for their computation. Since the adaptive agorithm depends on the whoe history of computed soutions, the time consumption is measured differenty for the uniform and the adaptive approach: Foruniformmesh-refinement, t unif, isthetimeeapsedforuniformmesh-refinements of the initia mesh T 0, the assemby of the Gaerkin data with respect to T, and the computation of the Gaerkin soution Φ with respect to T. For adaptive mesh-refinement, the computationa time is defined in an inductive manner: We define t adap, 1 := 0. For 0, t adap, is the sum of the previous steps t adap, 1 pus the time eapsed for the uniform refinement of T to obtain T, the assemby of the Gaerkin data with respect to T, the computation of the Gaerkin soution and the oca contributions of the error indicators, the marking step, and the oca refinement of T to obtain T Discussion of the numerica experiments. In Fig. 3 and Fig. 4, we compare the errors of approximations obtained by uniform mesh refinement and by use of the adaptive approach of Agorithm 14 with parameters θ = 0.4 and θ = 0.5, respectivey. We end up for T with 114,912 trianges after 30 adaptive refinement steps for θ = 0.4 and with 123,134 trianges after 26 adaptive refinement steps for θ = 0.5. In both cases, we observe a arge adaptivity ratio of h max /h min ,1 N 1/3 0,01 0,001 err unif, µ unif, osc unif, err adap, µ adap, osc adap, N 3/4 0, e+05 1e+06 number of eements Figure 3: Error pots for uniform and the adaptive refinement with parameter θ =

28 0,1 N 1/3 0,01 0,001 err unif, µ unif, osc unif, err adap, µ adap, osc adap, N 3/4 0, e+05 1e+06 number of eements Figure 4: Error pots for uniform and the adaptive refinement with parameter θ = 0.5. For the uniform refinement with up to 917,504 trianges, we observe a convergence of the energy error err of the Neumann data ike N 1/3, where N denotes the number of trianges as we as the number of degrees of freedom. This is in perfect agreement with the reguarity of the predescribed soution u H 5/3 (Ω), i.e., φ H 1/6 (Γ), and the a priori error estimate for uniform refinement φ Φ H 1/2 h s+1/2 φ H s for a possibe s [0,1]. For the data approximation error osc, we observe a higher order of convergence, as the Dirichet data g is zero at the reentrant edge and the adjacent faces. For the adaptive refinement, we observe a significanty higher order of convergence than for the uniform refinement. As we use an isotropic refinement to resove an edge singuarity, we cannot expect to get the optima convergence of N 3/4. But we observe an order of convergence of approximatey 0.7, i.e., cose to the optimum. The error estimator µ adap, proves to be a good estimator. In Fig. 5, we compare the errors err unif, and err adap, to the reated computationa times. On a first gance, the definition of t adap, and t unif, seems to favor uniform mesh refinement. However, we observe that the adaptive computations outperform the uniform computations significanty. Ony for sma computationa times and approximatey 10,000 uniform eements, the uniform refinement gives sighty smaer errors. Moreover, the computationa times are comparabe for both vaues of θ. Note that the computations in the adaptive agorithm do not take advantage of the fact that ony parts of the geometry are refined in each step and thus significant parts of the matrices coud be reused. Instead the matrices are generated from the scratch on each refinement eve. Therefore, cever impementation woud speed up the adaptive 28

29 0,1 0,1 uniform adaptive 0.4 adaptive 0.5 0,01 0,01 err_ 0,001 0,001 0,0001 0, computationa times in sec Figure 5: Errors err unif, and err adap, over computationa times for uniform refinement and adaptive refinement with parameter θ {0.4, 0.5}. computations significanty. In addition, the parameters of the FMM are chosen to be fixed during the whoe adaptive agorithm and are arge enough to guarantee no oss of accuracy for the finest eves. If we adjusted the parameters in a suitabe fashion to the actua error on each eve, we woud expect an additiona speedup of the computations. 7 Concusions 7.1. Anaytica Resuts. We proposed and anayzed an adaptive mesh-refinement agorithm for the numerica soution of the Lapace equation by a owest-order Gaerkinboundary eement method. To enabe the use of fast methods for boundary integra equations, we approximated the Dirichet data by discrete functions by means of the L 2 - projection onto piecewise inears. The resoution of the data approximation is incuded into the adaptive agorithm. This work transfers and extends the anaysis of [AFGKMP] to three dimensions. In the atter work, noda interpoation for the approximation of the Dirichet data is used. However, in the three-dimensiona case noda interpoation is not feasibeanymoreforh 1 -functions. WeproposetouseScott-Zhang-typequasi-interpoation operators or L 2 -orthogona projections. To that end, (oca) approximation estimates for quasi-interpoation operators in fractiona-order Soboev spaces were shown. We rigorousy prove that the proposed adaptive agorithm drives the error estimator to zero. The convergence of the computed discrete soutions to the exact soution in the energy norm φ Φ was shown under the so-caed saturation assumption Numerica Resuts. In the numerica experiment, the adaptive agorithm shows a 29

30 significanty higher order of convergence than the computations based on uniform meshrefinement. It amost regains the optima order even with the restriction to isotropic mesh-refinement, which cannot be optima for probems with generic edges singuarities; see [CMPS, Section 7.3]. Regarding the computationa error, the adaptive agorithm is faster than the computations with uniformy refined meshes, even though our BEM impementation is not adapted to the adaptive agorithm yet Future Work. For an adaptive agorithm driven by the weighted-residua error estimator from [CMS], even quasi-optimaity coud be proved recenty in [FKMP]. We aim to combine the ideas presented in the work at hand with those of [FKMP] to prove quasioptima convergence rates for adaptive agorithms driven by h h/2-based estimators. Furthermore, as aready mentioned, the impementation that was used for the experiments in this work was not fitted to the adaptive approach. Incuding the effective update of the system matrix into the adaptive agorithm, as was done e.g. in [DJ], wi certainy enhance computationa times. In addition, we aim to incude the choice of the parameters used for the Fast-Mutipoe method into the adaptive agorithm, thereby raising the accuracy that can be achieved by couping of the Fast-Mutipoe method and adaptivity with owest expenses. Acknowedgement The research of the authors M. Karkuik and D. Praetorius is supported through the FWF project Adaptive Boundary Eement Method, funded by the Austrian Science Fund (FWF) under grant P References [A] M. Ainsworth, J.T. Oden: A posteriori error estimation in finite eement anaysis, Wiey-Interscience [John Wiey & Sons], New York, [AFGKMP] M. Aurada, S. Ferraz-Leite, P. Godenits, M. Karkuik, M. Mayr, D. Praetorius: Convergence of adaptive BEM for some mixed boundary vaue probem, App. Numer. Math 62 (2012), [AFP] M. Aurada, M. Feisch, D. Praetorius: Convergence of some adatpvie FEM-BEM couping for eiptic but possiby noninear interface probems ESAIM: M2AN 46 (2012), [AFP+] M. Aurada, S. Ferraz-Leite, D. Praetorius: Estimator reduction and convergence of adaptive BEM, App. Numer. Math. 62 (2012), [AKP] M. Aurada, M. Karkuik, D. Praetorius: Simpe error estimators for hypersinguar integra equations in adaptive 3D-BEM, work in progress,