Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, Part I: Weakly-singular integral equation

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1 ASC Report No. 24/2013 Quai-optimal convergence rate for adaptive boundary element method with data approximation, Part I: Weakly-ingular integral equation M. Feichl, T. Führer, M. Karkulik, J.M. Melenk, and D. Praetoriu Intitute for Analyi and Scientific Computing Vienna Univerity of Technology TU Wien ISBN

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3 QUASI-OPTIMAL CONVERGENCE RATES FOR ADAPTIVE BOUNDARY ELEMENT METHODS WITH DATA APPROXIMATION, PART I: WEAKLY-SINGULAR INTEGRAL EQUATION M. FEISCHL, T. FÜHRER, M. KARKULIK, J. M. MELENK, AND D. PRAETORIUS Abtract. We analyze an adaptive boundary element method for Symm integral equation in 2D and 3D which incorporate the approximation of the Dirichlet data g into the adaptive cheme. We prove quai-optimal convergence rate for any H 1/2 -table projection ued for data approximation. 1. Introduction & Outline Data approximation i ubiquituou in numerical algorithm, and reliable, adaptive numerical cheme have to properly account for it. In thi direction, the preent work prove quai-optimal convergence rate for an adaptive boundary element method (ABEM) that include data error. A a model problem, we tudy Symm integral equation V φ = (1/2 + K)g on Γ := Ω (1) for given boundary data g H 1/2 (Γ) and a bounded Lipchitz domain Ω R d for d = 2, 3. The goal i to prove convergence and quai-optimality of ome tandard adaptive algorithm of the type olve etimate mark refine teered by a reidual-baed error etimator plu data approximation term. A focu of our analyi will be on the fact that the data g i not given exactly but approximated a part of the algorithm. Our theory cover everal of commonly ued technique to approximate g. In the framework of h-adaptive finite element method (AFEM) for econd order elliptic PDE, algorithm of thi type have been tudied in everal work, and convergence with quai-optimal algebraic rate can be proven (ee e.g. [BDD04, CKNS08, Dör96, FFP12, Ste07] and the reference therein). Naturally, one i intereted in the very ame quetion like convergence of the approximation and convergence rate alo for ABEM. The recent work [FKMP13] and [To13] lay out the path for proving quai-optimal convergence rate of ABEM with repect to the error etimator, or even for the energy error [AFF + 14]. However, the above mentioned work [AFF + 14, FKMP13, To13] are retricted to lowet-order dicretization and, more importantly, do not deal explicitly with data approximation. The preent paper raie ABEM to the ame level of mathematical undertanding a AFEM already i. More preciely, the improvement over the tate of the art are fourfold and read a follow: Firt, in contrat to the FEM, the right-hand ide in BEM typically involve boundary integral operator, which cannot be evaluated exactly in practice. Thu, the analyi of data error i mandatory. To compute the right-hand ide term (1/2 + K)g numerically Date: Augut 28, Mathematic Subject Claification. 65N30, 65N15, 65N38. Key word and phrae. boundary element method, weakly-ingular integral equation, a poteriori error etimate, adaptive algorithm, convergence, optimality. 1

4 in our model problem (1), we follow the earlier work [AFLG + 12, KOP13] and replace the exact data g by an approximate, piecewie polynomial data G l. Our approach thu decouple the problem of integrating the ingular kernel of the integral operator K from integrating the poibly ingular data g to compute Kg. On their own, both problem are well undertood and can be olved with tandard method. Moreover, in 2D (ee [Mai01]) one can even find analytic formula to compute the term KG l exactly. For d = 3, there alo exit black-box quadrature algorithm to compute KG l (ee, e.g., [SS11]). Second, in contrat to [FKMP13, To13], where only lowet-order BEM i conidered, the preent analyi work for arbitrary, but fixed-order dicretization. Third, we provide an improved analyi for the optimality of the Dörfler marking. Thi eliminate the efficiency contant in the etimate (even the weak efficiency ued in [FKMP13] i not needed) and in contrat to e.g. [CKNS08, FKMP13, To13] no lower error bound of any kind i involved. Finally, to deal with everal non-local data approximation term, we introduce a modified meh ize function. Thi may be of independent interet in the context of AFEM and ABEM ince it i pointwie equivalent to the uual meh ize function, but contractive not only on the refined element, but alo on an arbitrary but fixed number of element layer around them. An overall advantage of the preented approach i that the poible implementation only ha to deal with operator matrice of the dicrete integral operator. Thi i advantageou in term of fat boundary element method a e.g. H-matrice. Conequently, adaptive approximation of the Dirichlet data g eem to be the natural next tep to the final goal of a fat, fully dicrete, and black-box ABEM algorithm. Several other work deal with data approximation for adaptive BEM. However, they focu on convergence of the error etimator intead of proving quai-optimal algebraic convergence rate. In the 2D cae, [AFLG + 12] prove etimator convergence of ABEM for the Laplace problem with mixed boundary condition. The algorithm i teered by an (h h/2)-baed error etimator and alo approximate the given data adaptively by nodal interpolation. The work [KOP13] ue the L 2 -projection to prove etimator convergence of ABEM with data approximation in 3D. Both work do not guarantee any convergence rate of the etimator, and convergence of the error can only by proved under the aturation aumption, which i widely believed to hold true in practice, but till remain mathematically open for BEM (ee [AFF + 14] for the proof of a weaker form of thi aumption). In contrat to thi, the preent approach with reidual-baed error etimator guarantee convergence even with optimal rate. In particular, we prove that the optimal convergence i independent of the choen data approximation operator. The remainder of thi work i organized a follow: We preent the model problem a well a the adaptive algorithm for data approximation by mean of the Scott-Zhang projection for d = 2, 3 or nodal interpolation for d = 2 in Section 2. In Section 3, we develop ome crucial tool which are ued to prove convergence of the adaptive cheme in Section 4 and quai-optimality in Section 5. Boottrapping the foregoing reult, we are able to prove quai-optimal convergence of a lightly modified algorithm in Section 6 for each H 1/2 -table projection P l ued for data approximation, i.e. G l = P l g. Finally, Section 7 preent a numerical experiment which underline the reult of the work. Throughout the work, the ymbol abbreviate up to a multiplicative contant, and mean that both etimate and hold. Finally, #M denote the cardinality of a finite et M. 2

5 2. Model Problem & Adaptive Algorithm 2.1. Model problem. We conider Symm integral equation (1) where Γ := Ω i the boundary of a polygonal rep. polyhedral Lipchitz domain Ω R d, d = 2, 3. For d = 2, we enure diam(ω) < 1 by caling of the domain to guarantee the ellipticity of the implelayer operator V (ee (3) below). With n(x) R d denoting the exterior normal unit field at x Γ and the fundamental olution of the Laplacian 1 log z d = 2 G(z) := 2π for all z R d \ {0}, (2) 1 4π z 1 d = 3 the imple-layer operator V and the double-layer operator K formally read (V φ)(x) := G(x y)φ(y) dy and (Kg)(x) := p.v. n(y) G(x y)g(y) dy (3) Γ for all x Γ. Here, p.v. Γ denote Cauchy principal value. Then, (1) i an equivalent formulation of the Dirichlet problem u = 0 in Ω, u = g on Γ in the following ene [McL00]: The normal derivative φ = n u H 1/2 (Γ) of the olution u H 1 (Ω) of (4) olve (1) (Note that n u H 1/2 (Γ) i well-defined, ince u L 2 (Γ)). Converely, with φ H 1/2 (Γ) being a olution of (1), the repreentation formula u = V φ Kg H 1 (Ω) give the olution of (4), where the operator V and K are now evaluated in Ω intead of Γ Variational form and unique olvability. The operator V : H 1/2 (Γ) H 1/2 (Γ) i an elliptic and ymmetric iomorphim (ee e.g. the monograph [HW08, McL00, SS11]). It thu provide a calar product defined by φ, ψ := V φ, ψ L 2 (Γ). Thi calar product induce an equivalent norm on H 1/2 (Γ), which will be denoted by ψ := ψ, ψ 1/2. For ome Γ-dependent contant C norm > 0, it thu hold C 1 norm ψ ψ H 1/2 (Γ) C norm ψ for all ψ H 1/2 (Γ). (5) With thi, we may tate (1) equivalently a φ, ψ = (K )g, ψ for all ψ H 1/2 (Γ). (6) The fact that, i a calar product allow u to apply the Lax-Milgram lemma and hence guarantee exitence and uniquene of the olution φ H 1/2 (Γ) of (1). Wherea g H 1/2 (Γ) i ufficient to guarantee the olvability of (1), even without data approximation, it i neceary to require g H 1 (Γ) to formulate the weighted reidual error etimator η l. In the preent cae (ee Section 2.5 below), we make alo ue of the fact that we approximate an H 1 (Γ) function Sobolev pace and mapping propertie. Conider the Hilbert pace H 1 (Ω) := { v L 2 (Ω) : v L 2 (Ω) d}, equipped with the norm v 2 H 1 (Ω) := v 2 L 2 (Ω) + v 2 L 2 (Ω). We define the trace v Γ of a function v H 1 (Ω) by continuou extenion of the claical trace for mooth function. Thi permit the definition of 3 Γ (4)

6 the trace pace H 1/2 (Γ) := { v L 2 (Γ) : exit w H 1 (Ω) with v = w Γ } aociated with the norm v H 1/2 (Γ) := inf { w H 1 (Ω) : v = w Γ for w H 1 (Ω) } and it dual pace H 1/2 (Γ) := H 1/2 (Γ). We reviit the integral operator from (3) and continuouly extend them to the following boundary integral operator: V : H 1/2 (Γ) H 1/2 (Γ), K : H 1/2 (Γ) H 1/2 (Γ). According to [McL00, Chapter 7] there hold alo well-poedne and continuity of for all 1/2. V : H 1/2+ (Γ) H 1/2+ (Γ), K : H 1/2+ (Γ) H 1/2+ (Γ) Dicretization. Let T l denote a regular triangulation of Γ into compact and flat boundary implice T T l with Euclidean diameter diam(t ) and urface area T. Note that diam(t) = T for d = 2. For d = 3, we retrict to γ-hape regular mehe, i.e. γ 1 diam(t) T 1/2 γ diam(t) for all T T l (7) for a fixed contant γ 1. For d = 2, γ-hape regularity i undertood in the ene of max { diam(t)/diam(t ) : T, T T l and T T } γ. (8) Note that thi aumption doe not exclude trongly adapted mehe. Given a meh T l and p N {0}, we define the pace P p (T l ) := { Ψ l L 2 (Γ) : Ψ l T i a polynomial of degree at mot p on each T T l } of piecewie polynomial of degree p a well a the pace S p+1 (T l ) := { V l C(Γ) : V l T i a polynomial of degree at mot p + 1 on each T T l } of continuou pline function of degree p + 1. Note that there hold P p (T l ) H 1/2 (Γ) and S p+1 (T l ) H 1 (Γ). Moreover, we need generalized patche of ubet E l T l. We define the k-patch ω k l (E l ) T l inductively by k = 1 : ω l (E l ) := ω 1 l (E l) := { T T l : T T for ome T E l }, k > 1 : ω k l (E l) := ω l (ω k 1 l (E l )). Note that due to γ-hape regularity, there hold #E l #ωl k (E l ), where the hidden contant depend only on γ > 0 and k N, but not on E l or T l. Finally, for E l T l, we write E l := T E l T Γ. 4

7 2.5. Data approximation. To compute the right-hand ide (1/2 + K)g in the Galerkin cheme, we replace the exact Dirichlet data g by an approximation G l S p+1 (T l ). It remain to control the error introduced in thi way. To that end, we define the auxiliary olution φ l L 2 (Γ) H 1/2 (Γ) V φ l = (1/2 + K)G l. (9) The regularity of the olution φ l i a conequence of G l H 1 (Γ) and the mapping propertie of V 1 : H 1 (Γ) L 2 (Γ) and K : H 1 (Γ) H 1 (Γ). The problem we actually olve on the dicrete level read: Find Φ l P p (T l ) uch that Φ l, Ψ l = (1/2 + K)G l, Ψ l L 2 (Γ) (10) for all Ψ l P p (T l ). A for the continuou problem, the Lax-Milgram lemma applie and prove the unique olvability of (10). In Section 3 5, we analyze data approximation by mean of the Scott-Zhang projection J l : L 2 (Γ) S p+1 (T l ), i.e., G l := J l g. The mapping property J l : L 2 (Γ) S p+1 (T l ) deerve a comment. The original contruction of [SZ90] i defined on H 1 (Γ) o a to be able to conerve boundary condition. However, ince we are not intereted in conervation of boundary data (Γ ha no boundary), we may define the Scott-Zhang projection for a meh T l in an element-baed way on L 2 (Γ). We briefly ketch the contruction. We chooe a nodal bai of S p+1 (T l ) with bai function ξ z aociated with the Lagrangian node z N l of T l (i.e. ξ z (z ) = δ zz for all z, z N l and Kronecker delta δ zz {0, 1}). Note that upp(ξ z ) ω l (T z ) for ome arbitrary, but fixed element T z T l with z T z. Let ζ z P p+1 (T z ) denote the L 2 -dual bai function with repect to ξ z Tz, i.e. T z ξ z ζ z dx = δ zz for all z, z N l. Then, J l i defined a J l g := ( ) gζ z dx ξ z, T z z N l Obviouly, g L 2 (Γ) i ufficient to define J l g, and the reult of [SZ90] hold accordingly. Data approximation via other H 1/2 -table projection uch a the L 2 -orthogonal projection (for H 1 -tability ee [KPP13]) or the nodal interpolation for d = 2 i analyzed in Section Meh refinement. For local meh refinement, we ue the biection algorithm from [AFF + 14] for d = 2 and newet vertex biection, ee e.g. [Ver96, Chapter 4], for d = 3. For a et of marked element M l T l, we denote by T = refine(t l, M l ) the coaret refinement (with repect to the above mentioned biection algorithm), where at leat all element T M l are refined. To enure uniform γ-hape regularity (7) rep. (8), further refinement are made o that M l T l \ T. We write T refine(t l ) if there exit a finite equence of mehe T l,0 = T l, T l,1,..., T l,n = T and et of marked element M l,j T l,j for j = 0,..., N 1 uch that T l,j+1 := refine(t l,j, M l,j ) for all j = 0,..., N 1. The et of all mehe which can be obtained by refinement of the initial meh T 0 i denoted by T := {T refine(t 0 )}. (11) We emphaize the following two crucial propertie: Firt, the number of additional refinement which enure regularity and γ-hape regularity, doe not dominate the number 5

8 of marked element. More preciely, for T l = refine(t 0 ) with T j+1 = refine(t j, M j ) and M j T j for j = 0,..., l 1, it hold that l 1 #T l #T 0 C 1 #M j, (12) where C 1 > 0 depend only on T 0. Furthermore, for two mehe T, T l T, there exit a coaret common refinement T T l refine(t l ) refine(t ) uch that j=0 #(T T l ) #T + #T l #T 0. (13) Both propertie (12) and (13) are proved for d 2. The overlay etimate (13) wa firt proved for d = 2 in [Ste07] and then for d 2 in [CKNS08]. For newet vertex biection, the firt proof of (12) go back to [BDD04] for d = 2 and later [Ste08] for d 2. Their proof rely on a certain condition on the labelling of the reference edge in the initial meh T 0. For two-dimenional mehe, i.e. d = 3, the recent work [KPP13] proved that thi particular condition i not neceary and can be dropped. The biection algorithm for d = 2 i analyed in [AFF + 14] for d = 2. Moreover, uniform γ-hape regularity (7) rep. (8) hold for all mehe T l T with a contant γ > 0 which depend only on the initial meh T Error etimator. For error etimation, we employ the weighted reidual error etimator η l which date back to the eminal work [CS95, CS95, Car97] for 2D wa extended to 3D in [CMS01]. Given a meh T l a well a a olution Φ l of (10), the local contribution read η 2 l (T) := T 1/(d 1) (V Φ l (1/2 + K)G l ) 2 L 2 (T ) for all T T l. (14) Here, denote the urface gradient on Γ in the 3D cae. For 2D, reduce to the arc-length derivative along Γ. For any ubet E l T l, we write η 2 l (E l ) := T E l η 2 l (T). The global error etimator then read η l := η l (T l ) = h 1/2 l (V Φ l (1/2 + K)G l ) L 2 (Γ), where h l L (Γ) with h l T := T 1/(d 1) for d = 2, 3 denote the local meh width function. To control the error of the data approximation, we introduce the o called data ocillation term oc 2 l (T) := T 1/(d 1) (1 Π l ) g 2 L 2 (T ) for all T T l, (15) where Π l : L 2 (Γ) P p (T l ) denote the L 2 -orthogonal projection onto P p (T l ). For p = 0, thi i jut the piecewie integral mean. To abbreviate notation, we write oc 2 l (E l) = T E l oc 2 l(t) for all ubet E l T l and the global ocillation term i defined a oc l = oc l (T l ). The error etimator and the ocillation term are combined to ρ 2 l (T) := η2 l (T) + oc2 l (T) for all T T l. Again, we write ρ 2 l (E l) := T E l ρ 2 l (T) for each ubet E l T l and ρ 2 l = η2 l + oc2 l. 6

9 2.8. The adaptive algorithm. Now, we are able to tate the adaptive algorithm which will be proved to converge even with optimal rate. Algorithm 1. Input: initial meh T 0, adaptivity parameter 0 < θ < 1. Set l := 0 (i) Compute approximate data G l = J l g by ue of the Scott-Zhang projection. (ii) Compute olution Φ l of (10). (iii) Compute error etimator ρ l (T) for all T T l. (iv) Determine a et of marked element M l T l with minimal cardinality which atifie combined Dörfler marking θρ 2 l ρ2 l (M l). (16) (v) Refine at leat the marked element to obtain T l+1 = refine(t l, M l ). (vi) Increment l l + 1 and goto (i). Output: equence of error etimator (ρ l ) l N and equence of Galerkin olution (Φ l ) l N. Remark. To achieve the minimal cardinality of the et M l in tep (iv), one uually ort the quantitie ρ 2 l (T j) in decending order ρ l (T 1 ) ρ l (T 2 )... and define M l := {T 1,..., T j } for the minimal number j N uch that (16) i atified. In general, the et M l may not be unique. 3. Preliminarie Thi ection tate ome fact which are ued throughout the work. Firt, we hall need certain invere etimate. Propoition 2. Let T l T. Then, there exit a contant C inv > 0 uch that for all ψ L 2 (Γ) and v H 1 (Γ), it hold C 1 inv h1/2 l V ψ L 2 (Γ) ψ + h 1/2 l ψ L 2 (Γ), (17) C 1 inv h1/2 l (1/2 + K)v L 2 (Γ) v H 1/2 (Γ) + h 1/2 l v L 2 (Γ). (18) Particularly, for all Ψ l P p (T l ) and W l S p+1 (T l ), it hold that h 1/2 l Ψ l L 2 (Γ) + h 1/2 l V Ψ l L 2 (Γ) C inv Ψ l, (19) h 1/2 l W l L 2 (Γ) + h 1/2 l (1/2 + K)W l L 2 (Γ) C inv W l H 1/2 (Γ). (20) The contant C inv > 0 depend only on T 0 and p 0. Proof. The etimate (17) and (18) are proved in [AFF + 12, Theorem 1] rep. [Kar, Section 4.2]. The etimate (19) and (20) follow directly by employing the invere etimate from [GHS05, Theorem 3.6] for H 1/2 (Γ) h 1/2 l ( ) L 2 (Γ), and from [CP07, Corollary 3.2] for H 1/2 (Γ) h 1/2 l ( ) L 2 (Γ). The following lemma tate ome propertie of the Scott-Zhang projection. Lemma 3. Let T l T and g H 1 (Γ). For Γ denoting a (d 1)-dimenional manifold, the Scott-Zhang projection J l : L 2 (Γ) S p+1 (T l ) atifie h 1/2 l (1 J l )g L 2 (Γ) + (1 J l )g H 1/2 (Γ) C oc oc l. (21) Furthermore, for all refinement T refine(t l ), there hold the dicrete local upper bound (J J l )g H 1/2 (Γ) C oc oc l (ω 5 l (T l \ T )) (22) for all l N. The contant C oc > 0 depend only on T 0 and p 0. 7

10 Proof. The etimate (21) i a combination of [KOP13, Theorem 3] and [AFK + 13, Propoition 8], and the dicrete upper bound (22) i proved in [AFK + 13, Propoition 21]. Now, we take a cloer look at the error etimator ρ l. Propoition 4. The error etimator ρ l atifie (E1) Stability on non-refined element: There exit a contant C tab > 0 uch that C 1 ρ (T T l ) ρ l (T T l ) Φ Φ l + G G l H 1/2 (Γ) tab for all mehe T l, T T. (E2) Reduction property on refined element: There exit contant 0 < q red < 1 and C red > 0 uch that ρ 2 (T \ T l ) q red ρ 2 l (T ( ) l \ T ) + C red Φ Φ l 2 + G G l 2 H 1/2 (Γ) for all mehe T refine(t l ) with T l T. (E3) Reliability: There exit C rel > 0 uch that φ Φ l C rel ρ l for all mehe T l T. (E4) Dicrete local reliability: There exit a contant C dlr > 0 uch that Φ Φ l C dlr ρ l (R l ), (23) for all mehe T l T and refinement T refine(t l ) with correponding Galerkin olution Φ. Here R l := ω 5 l (T l \ T ) denote an extended et of refined element. Proof. We firt conider (E1). The fact that h l (Tl T ) = h (Tl T ) how oc l (T l T ) = oc (T l T ). Together with the triangle inequality, thi yield ρ (T T l ) ρ l (T T l ) h 1/2 V (Φ Φ l ) L 2 (Γ) + h 1/2 (1/2 + K)(G G l ) L 2 (Γ) Φ Φ l + G G l H 1/2 (Γ), where we have ued the invere inequalitie from Propoition 2 to get the final etimate. To ee (E2), we ue Young inequality with arbitrary δ > 0 and etimate ρ 2 (T \ T l ) (1 + δ) ( h 1/2 ( ) V Φ l (1/2 K)G l 2 L 2 ( T l \T + ) h1/2 (1 Π l ) g 2 L 2 ( T l \T ) + (1 + δ 1 )C inv ( Φ Φ l 2 + G G l 2 H 1/2 (Γ)), where we again applied the invere etimate from Propoition 2. Exploiting h T \T l 2 1/(d 1) h l Tl \T for d = 2, 3, we conclude the proof with q red = (1 + δ)2 1/(d 1) for ufficiently mall δ > 0. Reliability of η l with unperturbed right-hand ide i proved in [CS95, Theorem 2] for the 2D cae and in [CMS01, Corollary 4.3] for the 3D cae, i.e. To obtain (E3), we incorporate the data ocillation via φ l Φ l η l for all T l T. (24) φ Φ l φ φ l + φ l Φ l g G l H 1/2 (Γ) + η l, 8 )

11 where we ued the definition of φ l in (9) a well a the tability of V 1 : H 1/2 (Γ) H 1/2 (Γ) and K : H 1/2 (Γ) H 1/2 (Γ). Now, etimate (21) implie φ Φ l ρ l. Finally, dicrete local reliability (E4) of η l i proved in [FKMP13, Propoition 4.3] for p = 0. The proof hold verbatim for fixed p 0. Therefore, we get where Φ l Pp (T ) i the olution of Φ l Φ l η l (ω l (T l \ T )) η l (R l ), (25) Φ l, Ψ = (1/2 + K)G l, Ψ L 2 (Γ) for all Ψ P p (T ). We employ the definition of Φ l and the tability of Galerkin cheme to ee Φ l Φ G G l H 1/2 (Γ) oc l (R l ), where the lat etimate follow from (22). Finally, we combine the lat etimate with (25) and prove Thi conclude the proof. Φ Φ l Φ Φ l + Φl Φ l ρ l (R l ). Corollary 5. The error etimator i quai-monotone, i.e. for T refine(t l ) an arbitrary refinement of T l T, there hold ρ C mon ρ l with ome contant C mon > 0 which depend only on T 0 and p 0. Proof. The triangle inequality and h h l yield ρ ρ l + h 1/2 V (Φ Φ l ) L 2 (Γ) + h 1/2 (1/2 + K)(G G l ) L 2 (Γ) ρ l + Φ Φ l + G G l H (Γ), 1/2 where we applied the invere etimate (19) and (20) from Propoition 2 to obtain the lat etimate. Now, with dicrete local reliability (E4) and (22), we get Thi conclude the proof. ρ ρ l + ρ l (R l ) + oc l (R l ) ρ l. 4. Convergence of Algorithm 1 Thi ection analyze the convergence of Algorithm 1. Although the reult of thi ection may be intereting on their own (ince convergence of the adaptive algorithm i not clear a priori), they alo provide the crucial foundation for the optimality proof of Section Etimator reduction. The following etimator reduction reult i a very general concept and applie to numerou ituation in the context of a poteriori error etimation in BEM and FEM, ee e.g. [AFLP12] Propoition 6. Let T := refine(t l ) denote an arbitrary refinement of T l uch that the et of refined element T l \T atifie combined Dörfler marking (16) for ome 0 < θ < 1. Then, there exit contant 0 < q et < 1 and C et > 0 uch that ρ l atifie the perturbed contraction etimate ρ 2 q etρ 2 l + C et( Φ Φ l 2 + G G l 2 H 1/2 (Γ)). (26) The contant q et, C et > 0 depend only on θ, T 0, and p 0. 9

12 Proof. Recall Young inequality (a + b) 2 (1 + δ)a 2 + (1 + δ 1 )b 2 for all δ > 0 and a, b R. We exploit tability (E1) and reduction (E2) to ee ρ 2 = ρ 2 (T l T ) + ρ 2 (T \ T l ) (1 + δ)ρ 2 l (T l T ) + q red ρ 2 l (T l \ T ) + ( (1 + δ 1 )2C 2 tab + C red)( Φ Φ l 2 + G G l 2 H 1/2 (Γ)). Now, Dörfler marking (16) for T l \ T implie (1 + δ)ρ 2 l(t l T ) + q red ρ 2 l(t l \ T ) (1 + δ)ρ 2 l (1 + δ q red )ρ 2 l(t l \ T ) ( 1 + δ θ(1 + δ q red ) ) ρ 2 l. For ufficiently mall δ > 0, the combination of the lat two etimate prove (26) with q et = 1 + δ θ(1 + δ q red ) and C et = (1 + δ 1 )2C 2 tab + C red Contraction of quai-error. In thi ection, we make explicit ue of the fact that G l = J l g i obtained via the Scott-Zhang projection. A a theoretical tool, we introduce an equivalent meh width function that take care of the fact that in many of the local etimate below the patche of the element come into play. Lemma 7. Let k N be arbitrary and let (T l ) l N denote the equence of mehe generated by Algorithm 1. Then, there exit a modified meh width function h l uch that h l h l C 2 h l for all l N, (27) which i monotone and additionally provide a contraction on the k-patch of each refined element, i.e. for all l 1 it hold h l h l 1 pointwie almot everywhere in Ω, (28a) h l T q h l 1 T for all T ω k l (T l \ T l 1 ). (28b) The contant 0 < q < 1 and C 2 > 0 depend only on T 0 and on k N. Proof. Firt, we oberve that due to γ-hape regularity, the number of element in the k-patch i bounded, i.e. #ω k l (T) C 3 for all T T l. (29) The contant C 3 > 0 depend only on the γ-hape regularity and on k N. Recall the level function level( ) : l N T l N, which count the number of biection needed to generate an element T T l from it ancetor T T 0 T 0. By definition, there hold level(t 0 ) = 0 for all T 0 T 0 and for T T l \ T l 1, with father T T T l 1 there hold level(t) > level(t ). According to [KPP13, Lemma 18] the level difference of two neighbouring element T, T T l i le than or equal to two for d = 3 and bounded for d = 2 (ee [AFF + 14, Section 3]). Hence, the level difference of two element T, T T l which lie within one k-patch i alo bounded, i.e. level(t) level(t ) C 4 for T ωl k (T), (30) for ome contant C 4 > 0. We define a modified level-function inductively a follow: level 0 (T) := 0 for all T T 0 10

13 a well a for all l > 0 and all T T l level(t) T T l \ T l 1 level l (T) := level l 1 (T) + 1/(2C 4 C 3 + 1) T ωl k(t l \ T l 1 ) \ (T l \ T l 1 ) level l 1 (T) ele. Note that for T T l, the modified level level l (T) depend on the choen equence of mehe T 0,..., T l, in contrat to level(t). Below, we define the modified meh width function h l via the modified level function level l ( ). To obtain the equivalence (27), we firt how that the level function are equivalent, i.e. level(t) level l (T) level(t) + 2C 3 C 4 /(2C 3 C 4 + 1) for all l N and all T T l. (31) The lower bound follow from the definition of level l ( ) by induction on l. For the upper bound we argue by contradiction and aume the exitence of an element T T l with level l (T) > level(t) + 2C 3 C 4 /(2C 3 C 4 + 1). With the convention T 1 :=, let l 0 l be uch that T T l0 \ T l0 1. By definition of the modified level function, thi implie level l0 (T) = level(t) (obviouly, thi alo hold for l 0 = 0). Again, by definition of the modified level function, we know that the cae T ωl k j (T lj \ T lj 1) \ (T lj \ T lj 1) for l 0 < l j l mut have occurred at leat 2C 3 C time, ince otherwie level l (T) level(t) + 2C 3 C 4 /(2C 3 C 4 + 1). Put differently, ωl k j (T) (T lj \ T lj 1) for all thee (at leat 2C 3 C 4 + 1) many indice l 0 < l j l. Since ωl k 0 (T) contain at mot C 3 element, there i at leat one element T ωl k 0 (T) with T (T lj \ T lj 1) for at leat N max 2C 3C many indice l 0 < l j l. C 3 Hence, there exit an element T ωl k(t) with level(t ) N max +level(t ). Combining thi with (30), we obtain C 4 < C 4 + 1/C 3 N max C 4 level(t ) level(t ) level(t ) level(t) level(t ) level(t) C 4. Thi contradiction prove (31). Finally, with T 0 (T) T 0 denoting the unique father of T T 0 (T) in the initial meh T 0, we may define ( 1/(d 1) h l T := 2 levell (T ) T 0 (T) ) for all T T l and all l N. The property (27) follow immediately from the equivalence (31) a well a the fact that newet vertex biection for d 3 and imple biection for d = 2 guarantee ( 1/(d 1). T 1/(d 1) = 2 level(t ) T 0 (T) ) For (28b), we conider an element T ω k l (T l \ T l 1 ) with father T T T l 1. For T T, i.e. T T l \ T l 1, there hold level l (T) level(t) level(t ) + 1 level l 1 (T ) + 1/(2C 3 C 4 + 1), (32) 11

14 and therefore h l T h l 1 T ( 2 1/(2C 3 C 4 +1) ) 1/(d 1), (33) i.e. contraction in (28b) for T T l \ T l 1 with q = ( 2 1/(2C 3C 4 +1) ) 1/(d 1). For T = T, i.e. T ω k l (T l \ T l 1 ) \ (T l \ T l 1 ), we ee by definition of the modified level function that there alo hold level l (T) level l 1 (T ) + 1/(2C 3 C 4 + 1) which implie (33) and hence (28b) for T ωl k(t l\t l 1 )\(T l \T l 1 ). For T / ωl k(t l\t l 1 ), there hold level l 1 (T) = level l (T) and hence h l 1 T = h l T. Thi yield (28a) and conclude the proof. The next lemma provide a deciive improvement of our analyi compared to [AFK + 13]. Intead of treating all data approximation method with the technique of Section 6 below, we ue the locality of the Scott-Zhang projection together with the augmented contraction area of the modified meh width function h l, to prove a certain orthogonality relation of the approximate Dirichlet data. Thi allow u to ue the tandard Dörfler marking (16) in the adaptive algorithm (intead of the eparate Dörfler marking (62) a in Section 6) and i exploited in the proof of the contraction reult of Theorem 9. Lemma 8. Let T refine(t l ) denote a refinement of T l. Then, there hold C5 1 (J J l )g 2 1/2 H 1/2 (Γ) h l (1 Π l ) g 2 1/2 L 2 (Γ) h (1 Π ) g 2 L 2 (Γ) (34) for all l N. Here, h l denote the modified meh width function from Lemma 7 for k = 5. The contant C 5 > 0 depend only on T 0 and p 0. Proof. We employ Lemma 7 for k = 5 and obtain in combination with (22) (J J l )g H 1/2 (Γ) h 1/2 l (1 Π l ) g L 2 (ω 5 l (T l\t )) for all l N. (35) With monotonicity and the contraction property (28) of h l, we ee h l h (1 q) h l χ ω 5 l (T l \T ) for all l N, where χ ω 5 l (T l \T ) denote the characteritic function with repect to the et ωl 5 (T l \ T ). Together with (35), we therefore conclude (J J l )g 2 H 1/2 (Γ) ( h l h ) (1 Πl ) g 2 dx Γ = h 1/2 l (1 Π l ) g 2 L 2 (Γ) h 1/2 (1 Π l ) g 2 L 2 (Γ) h 1/2 l (1 Π l ) g 2 L 2 (Γ) according to the elementwie bet approximation property of Π. 1/2 h (1 Π ) g 2 L 2 (Γ) Theorem 9. Let T = refine(t l ) denote a refinement of T l T uch that T l \T atifie the combined Dörfler marking (16) (e.g. in Algorithm 1 with M l T l \ T l+1 ). Then, there exit contant 0 < α < 1, β > 0 and 0 < κ < 1 uch that the quai-error l := φ l Φ l 2 + αρ 2 l atifie the contraction property + β h 1/2 l (1 Π l ) g 2 L 2 (Γ) (36) κ l. Moreover, it hold αρ 2 l l (C7 2 + α + β)ρ2 l for all l N. The contant α, β, κ > 0 depend only on the ue of newet vertex biection, T 0, q et, and p 0. 12

15 Proof. Firt, we oberve that the data ocillation with the modified meh width function h l are till dominated by the error etimator, i.e. h 1/2 l (1 Π l ) g L 2 (Γ) oc l ρ l (37) for all l N. Second, we recall that tability of the problem allow u to control the influence of the data approximation in the ene of φ φ l 2 C 6 G G l 2 H 1/2 (Γ). (38) The contant C 6 > 0 depend only on the norm of V 1 and K. Furthermore, one ha reliability φ l Φ l C 7 η l C 7 ρ l (39) with C 7 > 0 independent of l N, cf. (24). Third, by definition of φ, ee (9), there hold orthogonality With (38), we infer φ Φ, Φ Φ l = 0. φ Φ 2 + Φ Φ l 2 = φ Φ l 2 (1 + δ) φ l Φ l 2 + (1 + δ 1 ) φ φ l 2 (1 + δ) φ l Φ l 2 + (1 + δ 1 )C 6 G G l 2 H 1/2 (Γ) for all δ > 0. Next, we ue the etimator reduction (26) to obtain (1 + δ) φ l Φ l 2 + (1 + δ 1 )C 6 G G l 2 H 1/2 (Γ) Φ Φ l 2 + αq et ρ 2 l + αc et ( Φ Φ l 2 + G G l 2 H 1/2 (Γ) (1 + δ) φ l Φ l 2 + ( (1 + δ 1 )C 6 + αc et ) G G l 2 H 1/2 (Γ) + αq et ρ 2 l + (αc et 1) Φ Φ l 2 + β h 1/2 (1 Π ) g 2 L 2 (Γ). With Lemma 8, thi implie for β = ( (1 + δ 1 )C 6 + αc et ) C5 (1 + δ) φ l Φ l 2 + (αc et 1) Φ Φ l 2 + αq et ρ 2 l + β h 1/2 l (1 Π l ) g 2 L 2 (Γ). Now, chooe αc et < 1 to implify the etimate above to (1 + δ) φ l Φ l 2 + αq et ρ 2 l (40) ) + β h 1/2 (1 Π ) g 2 L 2 (Γ) + β h 1/2 l (1 Π l ) g 2 L 2 (Γ). We introduce ome parameter ε > 0 and ue the bound (37) and (39) to obtain with (1 + δ C 1 7 ε) φ l Φ l 2 κ l + (αq et + 2ε)ρ 2 l 1/2 + (β ε) h l (1 Π l ) g 2 L 2 (Γ) κ := max{1 + δ C 1 7 ε, (αq et + 2ε)/α, (β ε)/β}. To enure 0 < κ < 1, chooe ε > 0 uch that αq et + 2ε < α and fix δ = C 1 7 ε/2. The equivalence ρ 2 l l follow immediately from (37) and (39). Thi prove the aertion. 13

16 5. Quai-Optimality of Algorithm 1 The quai-optimality proof roughly conit of two part: Firt, we prove in Propoition 10 that Dörfler marking (16) i not only ufficient for the contraction property tated in Theorem 9, but in ome ene even neceary. Second, Theorem 11 combine the foregoing reult with an etimate on the cardinality of the et of marked element M l and derive the optimality reult Optimality of marking criterion. The following propoition can be een a the convere of Theorem 9. Propoition 10. Let T refine(t l ) denote a refinement of T l. Let Φ l, Φ denote the olution of (10) correponding to G l = J l g, G = J g. Let 0 < κ < 1 and uppoe that the correponding error etimator atify ρ 2 κ ρ 2 l. (41) Then, there exit 0 < θ < 1 uch that ρ l atifie the combined Dörfler marking (16) for all 0 < θ θ with the et R l from (E4). θρ 2 l ρ2 l (R l) (42) Proof. Firt, we employ tability (E1) a well a the dicrete local reliability of the etimator (E4) and of the Scott-Zhang projection (22) to obtain, for arbitrary δ > 0, ρ 2 l = ρ2 l (T l \ T ) + ρ 2 l (T l T ) ρ 2 l (T l \ T ) + (1 + δ)ρ 2 + (1 + δ 1 )2C 2 tab( Φ Φ l 2 + (J J l )g 2 H 1/2 (Γ) (1 + (1 + δ 1 )2C 2 tab (C2 dlr + C2 oc ))ρ2 l (R l) + (1 + δ)κ ρ 2 l. We chooe δ > 0 uch that (1 + δ)κ < 1. Rearranging the term, we thu ee with θ ρ 2 l ρ2 l (R l) θ := (1 (1 + δ)κ )/(1 + (1 + δ 1 )2Ctab 2 (C2 dlr + C2 oc )) (0, 1). Thi conclude the proof of (42). Remark. Analyzing the proof of Theorem 11, we ee that we may chooe κ arbitrarily mall. Thi implie that for any 0 < θ < 1/(1+2Ctab 2 (C2 dlr +C2 oc )) := θ one may chooe δ > 0 ufficiently large and κ > 0 ufficiently mall uch that θ < θ Quai-optimal convergence rate. To conclude the quai-optimality proof in thi ection, we introduce the et of all mehe which have at mot N element more than the initial meh T 0, i.e. T N := { T T : #T #T 0 N } a well a the approximation cla A ρ,j (φ, g) A ρ,j def. characterized by (φ, g) A ρ,j := up min ρ N <. (43) T T N Here, φ and g are uppoed to be the olution and data of our model problem (1). Note that the method of data approximation G l g influence the error etimator ρ l. Hence, the definition of A ρ,j depend on the method of data approximation a well (which i hence indicated by the upercript ρ, J). Now, the quai-optimality i formulated in 14 N N )

17 the following theorem, which tate that each poible algebraic convergence rate for the error etimator will in fact be achieved by the ABEM algorithm. Theorem 11. For ufficiently mall parameter 0 < θ < 1, Algorithm 1 i optimal in the ene of (φ, g) A ρ,j ρ l C opt (#T l #T 0 ) for all l N. (44) The contant C opt > 0 depend only on (φ, g) A ρ,j, 0 < θ < 1, and on the contant 0 < κ < 1 from Theorem 9. Proof. Firt, Theorem 9 tate that l i a contractive equence, i.e. with 0 < κ < 1 given there l+1 κ l for all l N. (45) Now, let λ > 0 be a free parameter which i fixed later on. According to the definition of the approximation cla A ρ,j in (43), we find for ufficiently mall ε 2 := λρ 2 l > 0 ome triangulation T ε T uch that #T ε #T 0 ε 1/ and ρ ε ε, (46) where the hidden contant depend only on (φ, g) A ρ,j. We now conider the coaret common refinement T := T ε T l and firt note that #T #T l (#T ε + #T l #T 0 ) #T l = #T ε #T 0 (47) according to (13). Due to T refine(t ε ), we may apply Corollary 5 to get ρ 2 ρ2 ε λρ2 l. (48) Chooing λ > 0 ufficiently mall but fixed from now on, we enforce ρ 2 κ ρ 2 l for ome κ (0, 1) and ε ρ l. Next, we employ Propoition 10 to obtain that R l := ω 5 l (T l\t ) T l atifie the combined Dörfler marking (16). Recall that M l i choen in Step (iv) of Algorithm 1 to be a et with minimal cardinality. Together with the fact that each refinement plit the element into at leat two on, we get #M l #R l #(T l \ T ) #T #T l #T ε #T 0 ε 1/ ρ 1/ l. (49) Now, with optimality of the meh cloure (12) and contraction (45), we conclude l 1 #T l #T 0 #M l j=0 l 1 j=0 ρ 1/ j l 1 j=0 1/(2) j by convergence of the geometric erie. Finally, thi yield for all l N and conclude the proof. ρ l (#T l #T 0 ) 1/(2) l l 1 j=0 κ 1/(2) ρ 1/ l Remark. Reliability (E3) of the error etimator how the equivalence ρ 2 l ρ2 l + φ Φ l 2 for all l N. Therefore, the approximation cla A ρ,j can be equivalently defined in term of the total error, a i done in [CKNS08, FKMP13]. 15

18 5.3. Characterization of approximation cla. We aim to decouple the influence of the data approximation from the problem (1) in the approximation cla A ρ,j. To that end, we introduce φ A µ def. φ A µ := up N N min µ N <, (50) T T N where µ l denote the unperturbed etimator tudied in [FKMP13, AFF + 14] for adaptive BEM, i.e. µ l := h 1/2 l (V Φ µ l (1/2 + K)g) L 2 (Γ) with Φ µ l Pp (T l ) being the olution of the problem with exact right-hand ide Φ µ l, Ψ l = (1/2 + K)g, Ψ l L 2 (Γ) (51) for all Ψ l P p (T l ). Note that the definition of A µ doe not incorporate the data approximation G l g. Finally, we introduce the approximation cla g A oc def. g A oc := up min oc N <. (52) T T N The relation between the approximation clae introduced are dicued in the following theorem. Theorem 12. There hold the implication for all 1, 2 > 0. N N φ A µ 1, g A oc 2 = (φ, g) A ρ,j min{ 1, 2 }, (53) Proof. Firt, we ee by ue of (17) and (18) ρ 2 l µ2 l + h1/2 l (φ, g) A ρ,j = φ A µ, g Aoc (54) (1/2 + K)(g G l ) 2 L 2 (Γ) + h1/2 l V (Φ µ l Φ l) 2 L 2 (Γ) + oc2 l µ 2 l + g G l 2 H 1/2 (Γ) + h1/2 l (g G l ) 2 L 2 (Γ) + Φ µ l Φ l 2 + oc 2 l µ 2 l + oc 2 l for all T l T. Here, we ued the tability of Galerkin cheme a well a (21) to obtain the lat etimate. Analogouly, one prove the convere etimate to obtain ρ 2 l µ2 l + oc2 l. (55) Now, aume (φ, g) A ρ,j. For all N N, we obtain a meh T T N with N oc + N µ N ρ (φ, g) A ρ,j, where we ued (55). Thi implie immediately φ A µ + g A oc (φ, g) A ρ,j <, which prove φ A µ and g A oc and therefore (54). To ee (53), aume φ A µ 1 and g A oc 2 and define := min{ 1, 2 }. For all N N, the definition ofa µ 1 anda oc 2 yield mehe T µ and T oc with T µ, T oc T N/2 and µ µ (N/2) 1 φ A µ 1 and oc oc (N/2) 2 g A oc 2. 16

19 Now, we conider the coaret common refinement T := T µ T oc. Note that there hold #T #T 0 #T µ + #T oc 2#T 0 N due to (13). Moreover, we ee by ue of quai-monotonicity of the error etimator and the fact that oc oc oc Altogether, we ee ρ µ + oc µ µ + oc oc. ρ N µ µ N 1 + oc oc N 2 φ A µ + g A oc <. The hidden contant depend only C mon > 0 and the contant in Propoition 2. Thi prove (φ, g) A ρ,j and hence (53). Remark. For d = 2 and g H 2+ε (Γ) for ome ε > 0, the error etimator i even efficient up to term of higher order, i.e. and conequently µ 2 l φ Φ l 2 + hot 2 l, ρ 2 l φ Φ l 2 + oc 2 l + hot 2 l. The higher-order term atifie hot l h 3/2+ε for ome ε > 0 on quai-uniform mehe T l with meh width h = h l > 0 [AFF + 14, Theorem 4]. With [AFF + 14, Propoition 15] and analogou argument a in the proof above, one can characterize the approximation cla A µ for all 0 < 3/2 a where φ A def. φ A µ φ A and g A oc, φ A := up min N N T T N min φ Ψ N <. Ψ P 0 (T ) 6. Other Method of Data Approximation Thi ection analyze other method for approximating the Dirichlet data g. We ditinguih two cae for G l : For d = 2 and p = 0, i.e. Γ being a 1D manifold, we have g H 1 (Γ) C(Γ). Therefore, it i admiible to ue the nodal interpolation operator I l : C(Γ) S 1 (T l ) for data approximation. Each projection P l : H 1/2 (Γ) S p+1 (T l ) with l-independent tability contant i a valid choice. C P := up l N up v H 1/2 (Γ)\{0} P l v H 1/2 (Γ) v H 1/2 (Γ) < (56) The heart of the matter in the following ection i to compenate the lo of orthogonality (34) in the data approximation term. To thi end, we will ue a modified marking trategy from [Ste07], known a eparate Dörfler marking, which force the ocillation term to contract if it i big compared to the etimator. To clarify which method of data approximation i ued, we write e.g. Φ l,p for the olution of (10) and η l,p for the error etimator if the projection P l which i ued for data approximation i not the Scott-Zhang projection J l. 17

20 6.1. Data approximation by nodal interpolation G l = I l g for 2D BEM. The next lemma tate ome important propertie of I l. Note that throughout thi ection, we aume d = 2 and p = 0. Thi type of data approximation wa already conidered in [AFLG + 12] for the ymmetric BEM formulation of the 2D Laplace problem with inhomogeneou mixed boundary condition. However, in [AFLG + 12], only convergence of an (h h/2)-baed error etimator to zero i proved. Similar technique a employed here, are alo found in [FPP13], where the lowet-order AFEM of the 2D Laplace problem with inhomogeneou mixed boundary condition i conidered. Lemma 13. Let T l T. For a 1D manifold Γ, the nodal interpoland atifie C 1 oc (1 I l)g H 1/2 (Γ) h 1/2 l (1 I l )g L 2 (Γ) = oc l. (57) Moreover, it atifie the dicrete upper bound (I I l )g H 1/2 (Γ) C oc h 1/2 l (1 Π l ) g L 2 (T l \T ). (58) The contant C oc > 0 depend only on T 0. Proof. Etimate (57) follow from boottrapping the etimate (1 I l )v H 1/2 (Γ) h 1/2 l v L 2 (Γ) for all v H 1 (Γ) proved in [Car97, Theorem 1], ee e.g. [EFGP13, Lemma 2.2], and by ue of the wellknown identity I l v = Π l v L 2 (Γ) for all v H 1 (Γ) valid in 1D. Etimate (58) wa firt oberved in [FPP13, Proof of Propoition 3] and follow by imilar technique and I I l = (1 I l )I. By comparion with Lemma 3, we ee that the nodal interpolation operator I l ha the ame propertie a (and even tronger than) the Scott-Zhang projection J l. Thi implie that all the reult of the previou ection hold accordingly. In particular, the convergence reult of Theorem 9 remain valid. Moreover, we obtain the optimality reult of Theorem 11, if we replace the approximation cla A ρ,j with (φ, g) A ρ,i def. where the error etimator now read (φ, g) A ρ,i := up min ρ,p N <, (59) T T N N N ρ 2 l,i := h 1/2 l (V Φ l,i (1/2 + K)I l g) 2 L 2 (Γ) + oc 2 l. (60) 6.2. Data approximation by an H 1/2 -table projection G l = P l g for 2D and 3D BEM. General projection P l : H 1/2 (Γ) S p+1 (T l ) uually lack the dicrete upper bound (22). To overcome thi difficulty, we ue a lightly modified marking trategy known a eparate Dörfler marking [Ste07]. Thi variant wa alo ued in [AFK + 13] to prove quai-optimal convergence rate of AFEM for the Laplace problem with inhomogeneou Dirichlet data. Unlike [AFK + 13], we tre that our analyi of Section 4 above even cover the tandard Dörfler marking (16) if one ue the Scott-Zhang projection P l = J l for data approximation. In [KOP13], the cae G l = Π l g with Π l : L 2 (Γ) S p+1 (T l ) denoting the L 2 -orthogonal projection i conidered. According to [KPP13], newet vertex biection guarantee that Π l i H 1 (Γ) table and hence, by interpolation, alo H 1/2 (Γ) table. In contrat to the preent work, [KOP13] ue an (h h/2)-baed error etimator to teer the adaptive algorithm and prove only convergence of the etimator to zero without guaranteeing any convergence rate. 18

21 With the error etimator ρ 2 l,p = η2 l,p + oc2 l = h1/2 l (V Φ l,p (1/2 + K)P l g) 2 L 2 (Γ) + oc2 l (61) the adaptive algorithm now read a follow: Algorithm 14. Input: initial meh T 0, adaptivity parameter 0 < θ 1, θ 2, ϑ < 1. Set l := 0 (i) Compute approximate data G l = P l g. (ii) Compute olution Φ l,p of (10). (iii) Compute error etimator η l,p (T) and the data ocillation oc l (T) for all T T l. (iv) Determine a et of marked element M l T l with minimal cardinality which atifie eparate Dörfler marking: In cae of oc 2 l ϑη 2 l,p, find M l uch that θ 1 η 2 l,p η 2 l,p (M l ). In cae of oc 2 l > ϑη2 l,p, find M l uch that θ 2 oc 2 l oc2 l (M l). (62a) (62b) (v) Refine at leat the marked element to obtain T l+1 = refine(t l, M l ). (vi) Increment l l + 1 and goto (i). Output: equence of error etimator (ρ l,p ) l N and equence of Galerkin olution (Φ l,p ) l N. The next lemma how the equivalence of the olution and error etimator for different approximation of the Dirichlet data g up to the Dirichlet data ocillation oc l. Lemma 15. Let Φ l and Φ l,p denote olution of (10) correponding to the different approximation G l = J l g and G l = P l g of the Dirichlet data g. Then, it hold for any ubet E l T l Φ l Φ l,p C 8 oc l for all l N (63) a well a η l (E l ) η l,p (E l ) C 9 oc l for all l N, (64) where η l, η l,p denote the correponding error etimator from (14) rep. (61). The contant C 8 > 0 and C 9 > 0 depend only on T 0 and p 0. Proof. We tart with (63). By ue of tability of K : H 1/2 (Γ) H 1/2 (Γ), there hold Φ l Φ l,p (J l P l )g H 1/2 (Γ) g J l g H 1/2 (Γ) + g P l g H 1/2 (Γ). Now, we conclude with the H 1/2 -tability of P l g P l g H 1/2 (Γ) = (1 P l )(1 J l )g H 1/2 (Γ) (1 J l )g H 1/2 (Γ) oc 2 l, by ue of (21). The combination of the lat two inequalitie how (63). It remain to prove (64). To that end, we employ the triangle inequality a well a the invere etimate (19) (20) from Propoition 2 η l (E l ) η l,p (E l ) h 1/2 l V (Φ l Φ l,p ) L 2 (Γ) + h 1/2 l (1/2 + K)(J l P l )g L 2 (Γ) Φ l Φ l,p + (J l P l )g H (Γ). 1/2 Arguing a before to ee (J l P l )g H 1/2 (Γ) oc l, we conclude the proof. Now, we how that eparate Dörfler marking (62) for ρ 2 l,p = η2 l,p +oc2 l implie combined Dörfler marking (16) for ρ 2 l = η2 l + oc2 l. 19

22 Lemma 16. Let Φ l and Φ l,p denote olution of (10) correponding to different approximation G l = J l g and G l = P l g of the Dirichlet data g. Let the error etimator η l,p together with oc l atify the eparate Dörfler marking (62) for arbitrary 0 < θ 1, θ 2 < 1, ufficiently mall 0 < ϑ < 1, and a et of marked element M l T l. Then, ρ 2 l := η 2 l +oc 2 l atifie the combined Dörfler marking (16) for ome 0 < θ < 1. θρ 2 l ρ 2 l(m l ) (65) Proof. Firt, aume oc 2 l ϑη 2 l,p. Then, it hold with Lemma 15 η 2 l,p(m l ) 2η 2 l(m l ) + 2C 2 9oc 2 l 2η 2 l (M l ) + 2C 2 9ϑη 2 l,p. Moving the lat term to the left-hand ide, we ee (1 2C 2 9 ϑ)η2 l,p (M l) 2η 2 l (M l) 2ρ 2 l (M l). Together with Lemma 15 and (62a), thi yield θ 1 ρ 2 l 2θ 1η 2 l,p + θ 1(2C )oc2 l θ 1(2 + (2C )ϑ)η2 l,p (2 + (2C )ϑ)η2 l,p (M l) Second, aume oc 2 l > ϑη2 l,p. Then, again with Lemma 15 2(2 + (2C )ϑ)/(1 2C2 9 ϑ)ρ2 l (M l). (66) θ 2 ρ 2 l 2θ 2η 2 l,p + θ 2(2C )oc2 l θ 2(2ϑ 1 + 2C )oc2 l (2ϑ 1 + 2C )oc 2 l(m l ) (2ϑ 1 + 2C )ρ 2 l(m l ). Hence, ρ l atifie combined Dörfler marking (16) with (67) θ := min{θ 1 (1 2C 2 9ϑ)/(4 + 2(2C )ϑ), θ 2 /(2ϑ 1 + 2C )}. (68) Thi conclude the proof. Remark. We etablihed convergence of Algorithm 14 at leat for ufficiently mall 0 < ϑ < 1. The previou lemma how that ρ l atifie combined Dörfler marking (16) in each tep of the adaptive loop. Therefore, Theorem 9 i applicable and implie ρ 2 l l 0 a l. The equivalence ρ l,p ρ l which follow immediately from (64) prove lim l ρ l,p = 0. Remark. Arguing a in the proof of Lemma 16, we ee that eparate Dörfler marking (62) for ρ l,p alo implie combined Dörfler marking (16) for ρ l,p without any aumption on 0 < θ 1, θ 2, ϑ < 1. Then, the etimator reduction (26) of Propoition 6 alo hold in thi cae. In [KOP13], it i proved that thi implie convergence of the adaptive algorithm. Lemma 17. The error etimator ρ l,p atifie (E1) (E3) and there hold quai-monotonicity, i.e. for T refine(t l ) being a refinement of T l T, we have for a contant C 10 > 0 which depend only on T 0. ρ,p C 10 ρ l,p (69) Proof. (E1) (E3) follow verbatim a in the proof of Propoition 4. For quai-monotonicity, we apply the equivalence (64) to obtain ρ,p ρ ρ l ρ l,p. Thi conclude the proof. 20

23 Propoition 18. Let T refine(t l ) be a refinement of T l T. Let Φ l,p, Φ,P denote the correponding olution of (10). Suppoe that the error etimator atifie ρ,p κ ρ l,p (70) for ome 0 < κ < 1 ufficiently mall. Then, the et R l := ω 5 l (T l \ T ) from (E4) atifie eparate Dörfler marking (62) for ufficiently mall 0 < θ 1, ϑ < 1 and arbitrary 0 < θ 2 < 1. Proof. Firt, we prove by ue of (64) ρ 2(C 9 + 1)ρ,P 2(C 9 + 1)κ ρ l,p 2(C 9 + 1) 2 κ ρ l, (71) Therefore and with 2(C 9 + 1) 2 κ < 1, we may apply Propoition 10 to conclude θ ρ 2 l ρ 2 l(r l ). (72) Now, we ditinguih two cae. Firt, aume oc 2 l ϑη 2 l,p. Then, it hold with (64) η 2 l,p 2η2 l + 2C2 9 oc2 l 2η2 l + 2C2 9 ϑη2 l,p. Now, we rearrange the term in the above equation and employ (72) to get (1 2C 2 9 ϑ)θ η 2 l,p 2η2 l (R l) + 2oc 2 l (R l) 4η 2 l,p (R l) + (2C )ϑη2 l,p. For ϑ > 0 ufficiently mall, thi how ( (1 2C 2 9 ϑ)θ (2 + 2C 2 9 )ϑ) /4 η 2 l,p η2 l,p (R l), i.e. (62a) for all θ 1 ((1 2C 2 9 ϑ)θ (2 + 2C 2 9 )ϑ)/4. Second, let oc 2 l > ϑη2 l,p. Here, we ue the local definition of oc l to obtain Now, we conclude oc 2 l(t l T ) = h 1/2 l (1 Π l ) g L 2 ( T l T ) = oc 2 (T l T ) ρ 2 2(C 9 + 1) 2 κ ρ 2 l 2(C 9 + 1) 2 κ (2η 2 l,p + (1 + 2C 2 9)oc 2 l) 2(C 9 + 1) 2 κ (2ϑ C 2 9)oc 2 l. oc 2 l = oc 2 l(t l T ) + oc 2 l(t l \ T ) 2(C 9 + 1) 2 κ (2ϑ C 2 9)oc 2 l + oc 2 l(r l ), which how (62b) for all θ 2 1 2(C 9 + 1) 2 κ (2ϑ C9 2 ). Moreover, for all 0 < θ 2 < 1 we may chooe κ uch that (62b) hold true. Thi conclude the proof. Now, we have collected all ingredient to prove an optimality reult imilar to Theorem 11. To that end, we define the approximation cla (φ, g) A ρ,p def. (φ, g) A ρ,p := up min ρ l,p N <. (73) T T N Theorem 19. For ufficiently mall parameter 0 < θ 1, ϑ < 1 and arbitrary 0 < θ 2 < 1, Algorithm 14 i optimal in the ene of (φ, g) A ρ,p ρ l,p C 11 (#T l #T 0 ) for all l N. (74) The contant C 11 > 0 depend only on (φ, g) A ρ,p, 0 < θ 1, θ 2, ϑ < 1, and 0 < κ < N N