M. Aurada 1,M.Feischl 1, J. Kemetmüller 1,M.Page 1 and D. Praetorius 1

Transcription

1 ESAIM: M2AN 47 (2013) DOI: /m2an/ ESAIM: Mathematica Modeing and Numerica Anaysis EACH H 1/2 STABLE PROJECTION YIELDS CONVERGENCE AND QUASI OPTIMALITY OF ADAPTIVE FEM WITH INHOMOGENEOUS DIRICHLET DATA IN R d M. Aurada 1,M.Feisch 1, J. Kemetmüer 1,M.Page 1 and D. Praetorius 1 Abstract. We consider the soution of second order eiptic PDEs in R d with inhomogeneous Dirichet data by means of an h adaptive FEM with fixed poynomia order p N. As mode exampe serves the Poisson equation with mixed Dirichet Neumann boundary conditions, where the inhomogeneous Dirichet data are discretized by use of an H 1/2 stabe projection, for instance, the L 2 projection for p = 1 or the Scott Zhang projection for genera p 1. For error estimation, we use a residua error estimator which incudes the Dirichet data osciations. We prove that each H 1/2 stabe projection yieds convergence of the adaptive agorithm even with quasi optima convergence rate. Numerica experiments with the Scott Zhang projection concude the work. Mathematics Subject Cassification. 65N30, 65N50. Received February 23, Revised September 22, Pubished onine June 17, Introduction Recenty, there has been a major breakthrough in the thorough mathematica understanding of convergence and quasi optimaity of h adaptive FEM for second order eiptic PDEs. However, the focus of the numerica anaysis usuay ies on mode probems with homogeneous Dirichet conditions, i.e. Δu = f in Ω with u =0 on Γ = Ω, seee.g. [5, 10 12, 17, 21, 27]. Instead, our mode probem Δu = f in Ω, u = g on Γ D, n u = φ on Γ N (1.1) considers inhomogeneous mixed Dirichet Neumann boundary conditions. Here, Ω is a bounded Lipschitz domain in R d with poyhedra boundary Γ = Ω which is spit into two (possiby non connected) reativey open boundary parts, namey the Dirichet boundary Γ D and the Neumann boundary Γ N, i.e. Γ D Γ N = and Keywords and phrases. Adaptive finite eement method, convergence anaysis, quasi optimaity, inhomogeneous Dirichet data. 1 Institute for Anaysis and Scientific Computing, Vienna University of Technoogy, Wiedner Hauptstraße 8-10, 1040 Wien, Austria. {Markus.Aurada, Michae.Feisch, Josef.Kemetmueer, Dirk.Praetorius}@tuwien.ac.at; Marcus.Page@tuwien.ac.at Artice pubished by EDP Sciences c EDP Sciences, SMAI 2013

2 1208 M. AURADA ET AL. Γ D Γ N = Γ. We stress that the surface measure of the Dirichet boundary has to be positive Γ D > 0, whereas Γ N is aowed to be empty. The given data formay satisfy f H 1 (Ω), g H 1/2 (Γ D ), and φ H 1/2 (Γ N ). We refer to Section 2.6 beow for the definition of these Soboev spaces. As is usuay required to derive (ocaized) a posteriori error estimators, we assume additiona reguarity of the given data, namey f L 2 (Ω), g H 1 (Γ D ), and φ L 2 (Γ N ). Moreover, we assume that the boundary partition into Γ D and Γ N is resoved by the trianguations used. We stress that using resuts avaiabe in the iterature it is easiy possibe to generaize the anaysis from the Lapacian L = Δ to genera uniformy eiptic differentia operators of second order. The reader is referred to the semina work [10] which treats the case of Γ D = Ω and homogeneous Dirichet data g =0andprovides the anaytica toos to cover genera symmetric L, whie the recent work [11] extends this anaysis to non symmetric L. Therefore, we ony focus on the nove techniques which are necessary to dea with inhomogeneous Dirichet data. Unike the case g = 0 which is we studied in the iterature, see e.g. [3, 30], ony itte work has been done on a posteriori error estimation for (1.1) withg 0,cf. [4, 24]. Moreover, besides the 2D works [14, 22], no convergence resut for AFEM with inhomogeneous Dirichet data is found in the iterature, yet. Whie the incusion of inhomogeneous Neumann conditions φ into the convergence anaysis of e.g. [5, 10 12, 17, 21, 27] is straightforward, incorporating inhomogeneous Dirichet conditions g is not obvious and technicay much more demanding for severa reasons: First, since discrete FE functions cannot satisfy genera inhomogeneous Dirichet conditions g, the FE scheme requires an additiona discretization of g g. Second, the error g g H 1/2 (Γ D) of this data approximation has to be controed with respect to the non oca H 1/2 norm and has to be incuded in the a posteriori error anaysis and the adaptive agorithm. Third, in contrast to the case g = 0, the discrete ansatz spaces V are non-nested, i.e. V V +1. We therefore oose the orthogonaity in energy norm which eads to certain technicaities to construct a contraction quantity which is equivaent to the Gaerkin error resp. error estimator. Therefore, quasi optimaity as we as even pain convergence of AFEM with inhomogeneous Dirichet data is not obvious at a. The earier works [14, 22] considered owest order finite eements p = 1 in 2D and noda interpoation to discretize g. However, this situation is very specia in the sense that the entire anaysis in [14, 22] is stricty bound to the owest order case and cannot be generaized to R d, since noda interpoation of the Dirichet data is we-defined if and ony if d =2.[22] usedanerrorestimatorwhichisobtainedbysovingocaprobems on stars, and proved convergence of the reated adaptive agorithm. This convergence resut, however, reies on an artificia marking criterion which first consists of Dörfer marking [12] for the estimator, and afterwards some possibe enrichment of the set of marked eements to guarantee inear convergence of voume and Dirichet osciations. In [14], the common residua-based error estimator is anayzed, and combined marking for estimator pus Dirichet osciations is shown to ead to convergence and even quasi optimaity of AFEM. In this work, we consider finite eements of piecewise poynomia order p 1 and dimension d 2. We show that each uniformy H 1/2 (Γ D ) stabe projection P onto the discrete trace space wi do the job: In this frame, we may use techniques from adaptive boundary eement methods [9, 13, 18] to ocaize the non oca H 1/2 norm in terms of a ocay weighted H 1 seminorm. To overcome the ack of Gaerkin orthogonaity, the remedy is to concentrate on a quasi Pythagoras theorem and a stronger marking criterion. The atter is used to guarantee (quasi-oca) equivaence of error estimators for different discretizations of the Dirichet data. To obtain contraction of our AFEM, we may then consider (theoreticay) the H 1/2 (Γ D ) orthogona projection. To obtain optimaity of the marking strategy, we may consider the Scott Zhang projection instead. Both auxiiary probems are somehow sufficienty cose to the origina probem with projection P, which is enforced by the marking strategy. Overa, we prove that each uniformy H 1/2 stabe projection P wi ead to a convergent AFEM agorithm. Under the usua restrictions on the adaptivity parameters, we even show optima agebraic convergence behaviour with respect to the number of eements.

3 AFEM WITH INHOMOGENEOUS DIRICHLET DATA Adaptive agorithm It is we-known that the Poisson probem (1.1) admits a unique weak soution u H 1 (Ω) withu = g on Γ D in the sense of traces which soves the variationa formuation u, v Ω = f,v Ω + φ, v ΓN for a v HD 1 (Ω). (2.1) Here, the test space reads H 1 D (Ω) ={ v H 1 (Ω) : v =0onΓ D in the sense of traces },and, denotes the respective L 2 scaar products. The proof reies essentiay on a reformuation of (1.1) as a probem with homogeneous Dirichet data via a so caed ifting operator L, i.e. L : H 1/2 (Γ ) H 1 (Ω) is a inear and continuous operator with (Lĝ) Γ = ĝ for a ĝ H 1/2 (Γ ) in the sense of traces. Again, we refer to Section 2.6 for the definition of the trace space H 1/2 (Γ ). However, athough L is constructed anayticay, it is hardy accessibe numericay in genera and thus this approach is not feasibe in practice. This section provides an overview on this work and its main resuts. We anayze a common adaptive mesh refining agorithm of the type sove estimate mark refine which is stated in detai beow in Section 2.5. We start with a discussion of its four modues The modue sove Let T be a reguar trianguation of Ω into simpices, i.e. tetrahedra for 3D resp. trianges for 2D, which is generated from an initia trianguation T 0.LetE be the set of facets, i.e. faces for 3D and edges for 2D, respectivey. This set is spit into interior facets E Ω = { E E : E Ω }, i.e. each E E Ω satisfies E = T + T for T ± T, as we as boundary facets E Γ = E \E Ω. We assume that the partition of Γ into Dirichet boundary Γ D and Neumann boundary Γ N is aready resoved by the initia mesh T 0, i.e. E Γ is spit into E D = { } E E : E Γ D and E N = { } E E : E Γ N for a 0. Note that E D (resp. E N ) therefore provides a reguar trianguation of the boundary Γ D (resp. Γ N ). We use conforming eements of fixed poynomia order p N. ByP p (T ) we denote the space of poynomias of degree p on T T,andby P p (T )= { V : Ω R : V T P p } (T ) for a T T (2.2) the space of eementwise poynomias. The corresponding FEM ansatz space then reads S p (T )= { V C(Ω) :V T P p (T ) for a T T }. (2.3) Since a discrete function U S p (T ) cannot satisfy genera continuous Dirichet conditions, we have to discretize the given data g H 1 (Γ D ). To this purpose, et P : H 1/2 (Γ D ) S p (E D ) be a projection onto the discrete trace space S p (E D )={ V ΓD : V S p (T ) }. (2.4) As in the continuous case, it is we-known that there is a unique U S p (T )withu = P g on Γ D which soves the Gaerkin formuation U, V Ω = f,v Ω + φ, V ΓN for a V S p D (T ). (2.5) Here, the test space is given by S p D (T )=S p (T ) H 1 D (Ω) ={ V S p (T ):V =0onΓ D }. We assume that sove computes the exact Gaerkin soution of (2.5). Arguing as e.g. in [7,27], it is, however, possibe to incude an approximate sover into our anaysis. Possibe choices for P incude the L 2 orthogona projection for the owest order case p = 1, which is considered in [4], or the Scott Zhang projection from [26] which is proposed in [24].

4 1210 M. AURADA ET AL The modue estimate We start with the eement data osciations osc 2 T, := T T osc T, (T ) 2, where osc T, (T ) 2 := T 2/d (1 Π T )(f + ΔU ) 2 L 2 (T ) (2.6) and where Π T : L 2 (T ) P p 1 (T ) denotes the L 2 orthogona projection. These arise in the efficiency estimate for residua error estimators. Moreover, the efficiency invoves the Neumann data osciations osc 2 N, := osc N, (E) 2, where osc N, (E) 2 := T 1/d (1 Π )φ 2 L 2 (E) (2.7) E E N with T T being the unique eement with E T and where Π : L 2 (Γ ) P p 1 (E Γ ):={ W Γ : W P p 1 (T ) } denotes the (E Γ piecewise) L 2 orthogona projection on the boundary. Finay, the approximation of the Dirichet data P g g H 1 (Γ D ) is controed by the Dirichet data osciations osc 2 D, := osc D, (E) 2, where osc D, (E) 2 := T 1/d (1 Π ) Γ g 2 L 2 (E), (2.8) E E D where again T T denotes the unique eement with E T. Moreover, Γ ( ) denotes the surface gradient. We reca that up to shape reguarity we have equivaence T 1/d diam(t )asweasdiam(t) diam(e) for a T T and E E with E T. We use a residua error estimator η 2 = ϱ2 +osc2 D, which is spit into genera contributions and Dirichet osciations, i.e. ϱ 2 = ϱ (T ) 2 (2.9) T T with corresponding refinement indicators ϱ (T ) 2 := T 2/d f + ΔU 2 L 2 (T ) + T 1/d( [ n U ] 2 L 2 ( T Ω) + φ nu 2 L 2 ( T Γ N )). (2.10) The modue estimate returns the eementwise contributions ϱ (T ) 2 and osc D, (E) 2 for a T T and E E D The modue mark For eement marking, we use a modification of the Dörfer marking [12] proposed firsty in Stevenson [27]. In each step of the adaptive oop, we mark either eements or Dirichet facets for refinement, where the atter is ony done if osc D, is arge when compared to ϱ. A precise statement of the modue mark is part of Agorithm 2.1 beow The modue refine Locay refined meshes are obtained by use of the newest vertex bisection agorithm, see e.g. [28, 29], where T +1 = refine(t, M )forasetm T of marked eements returns the coarsest reguar trianguation T +1 such that a marked eements T M have been refined by at east one bisection. Arguing as in e.g. [17], one may aso use variants of newest vertex bisection, where each T M is refined by at east n bisections with arbitrary, but fixed n N.

5 2.5. Adaptive oop AFEM WITH INHOMOGENEOUS DIRICHLET DATA 1211 With the aforegoing modues, the adaptive mesh refining agorithm takes the foowing form. Agorithm 2.1. Let adaptivity parameters 0 <θ 1,θ 2,ϑ < 1 and initia trianguation T 0 be given. For each =0, 1, 2,... do: (i) Compute discrete soution U S p (T ). (ii) Compute refinement indicators ϱ (T ) and osc D, (E) for a T T and E E D. (iii) Provided that osc 2 D, ϑϱ2, choose M T such that θ 1 ϱ 2 (iv) Provided that osc 2 D, >ϑϱ2, choose MD ED such that θ 2 osc 2 D, T M ϱ (T ) 2. (2.11) E M D and et M := { T T : E M D E T }. (v) Use newest vertex bisection to generate T +1 = refine(t, M ). (vi) Update counter +1 andgoto(i). osc D, (E) 2 (2.12) Remark 2.2. The modified Dörfer marking in step (iii) (iv) of the above agorithm which first appeared in [27], is stronger than the usua Dörfer marking [12], see (4.1) (4.2) beow, which is used in other works on AFEM for homogeneous Dirichet data g =0,cf. e.g. [10,11,17]. This is, however, a necessity since our anaysis expoits properties of different discretizations g = P g of g for our proof of inear convergence (Thm. 2.6) and quasi optimaity (Thm. 2.8). In the proofs, we use that the modified marking strategy for an estimator corresponding to a discretization g impies the usua Dörfer marking for an estimator corresponding to another discretization, cf. Lemma 5.2. Note, however, that pain convergence of AFEM (Thm. 2.5) sti hods for the usua Dörfer marking (4.1) (4.2). Moreover, in the 2D case, where the Dirichet data are discretized by means of noda interpoation, inear convergence and even quasi optimaity can be shown for the usua Dörfer marking due to some additiona orthogonaity reation of 1D noda interpoation, cf. [14] Function spaces This section briefy coects the function spaces and norms used in the foowing.werefere.g. to the monographs [16,20,25] for further detais. In particuar, detais on the fractiona order Soboev spaces used throughout can be found here. L 2 (Ω) resp.h 1 (Ω) denote the usua Lebesgue space and Soboev space on Ω. The dua space of H 1 (Ω) with respect to the extended L 2 (Ω) scaar product is denoted by H 1 (Ω). For measurabe γ Γ, e.g. γ {Γ D,Γ N }, the Soboev space H 1 (γ) is defined as the competion of the Lipschitz continuous functions on γ with respect to the norm v 2 H 1 (γ) = v 2 L 2 (γ) + Γ v 2 L 2 (γ),where Γ ( ) denotes the surface gradient for d = 3 resp. the arcength derivative for d = 2. With the Lebesgue space L 2 (γ), Soboev spaces of fractiona order 0 α 1 are defined by interpoation H α (γ) =[L 2 (γ); H 1 (γ)] α.moreover, H 1 (γ) is defined as the competion of a Lipschitz continuous functions on γ which vanish on γ, with respect to the H 1 (γ) norm, and H α (γ) =[L 2 (γ); H 1 (γ)] α is defined by interpoation. Soboev spaces of negative order are defined by duaity H α (γ) =H α (γ) and H α (γ) = H α (γ),where duaity is understood with respect to the extended L 2 (γ) scaar product. We note that H 1/2 (Γ ) can equivaenty be defined as the trace space of H 1 (Ω), i.e. H 1/2 (Γ )= { ŵ Γ : ŵ H 1 (Ω) }. (2.13)

6 1212 M. AURADA ET AL. For our anaysis, we sha use the graph norm of the restriction operator w H 1/2 (Γ ) := inf { ŵ H1 (Ω) : ŵ H 1 (Ω) withŵ Γ = w }. (2.14) Moreover, the graph norm and the interpoation norm are, in fact, equivaent norms on H 1/2 (Γ ), and the norm equivaence constants depend ony on Γ. A simiar observation hods for the space H 1/2 (γ), namey and the corresponding graph norm H 1/2 (γ) = { ŵ γ : ŵ H 1/2 (Γ ) }, (2.15) w H 1/2 (γ) := inf { ŵ H 1/2 (Γ ) : ŵ H 1/2 (Γ )withŵ γ = w } (2.16) are equivaent norms on H 1/2 (γ). Throughout our anaysis and without oss of generaization, we sha equip H 1/2 (Γ )resp.h 1/2 (γ) with these graph norms (2.13) (2.16) Main resuts Throughout, we assume that the projections P : H 1/2 (Γ D ) S p (E D ) are uniformy H1/2 (Γ D ) stabe, i.e. the operator norm is uniformy bounded P : H 1/2 (Γ D ) H 1/2 (Γ D ) C stab < (2.17) with some independent constant C stab > 0. This assumption is guaranteed for the H 1/2 (Γ D ) orthogona projection with C stab =1.Moreover,theL 2 (Γ D ) orthogona projection for the owest order case p =1and newest vertex bisection is uniformy bounded [19], and so is the Scott Zhang projection [26] ontos p (E D)for arbitrary p 1. First, our discretization is quasi optima in the sense of the Céa emma. Note that estimate (2.18) doesnot depend on the precise choice of P, and the minimum is taken over a discrete functions. Unike our observation, the resut in e.g. [4], Theorem 6.1 takes the minimum with respect to the affine space { W S p (T ):W ΓD = P g } and for first order p =1ony. Proposition 2.3 (Céa type estimate in H 1 norm). The Gaerkin soution satisfies u U H1 (Ω) C Céa min u W H1 (Ω). (2.18) W S p (T ) The constant C Céa > 0 depends ony on Ω, Γ D, shape reguarity of T, the poynomia degree p 1, andthe constant C stab > 0. Second, the considered error estimator provides an upper bound and, up to data osciations, aso a ower bound for the Gaerkin error. Proposition 2.4 (reiabiity and efficiency of η ). The error estimator η 2 = ϱ2 +osc2 D, is reiabe u U 2 H 1 (Ω) C re η 2 (2.19) and efficient C 1 eff η2 (u U ) 2 L 2 (Ω) +osc2 T, +osc 2 N, +osc 2 D,. (2.20) The constants C re,c eff > 0 depend on Ω and Γ D, on the poynomia degree p 1, stabiity C stab > 0, the initia trianguation T 0, and on the use of newest vertex bisection.

7 AFEM WITH INHOMOGENEOUS DIRICHLET DATA 1213 Note that convergence of Agorithm 2.1 in the sense of im U = u in H 1 (Ω) isaprioriuncear since adaptive mesh refinement does not guarantee that the oca mesh size tends to zero. However, we have the foowing convergence resut which is proved in the frame of the estimator reduction concept from [2]. Theorem 2.5 (convergence of AFEM). (i) Suppose that the discretization of the Dirichet data guarantees some aprioriconvergence im g P g H 1/2 (Γ D) = 0 (2.21) with a certain (yet unknown) imit g H 1/2 (Γ D ). Then, for any choice of the adaptivity parameters 0 <θ 1,θ 2,ϑ<1, Agorithm 2.1 guarantees convergence im u U H 1 (Ω) = 0 (2.22) and, in particuar, g = g. (ii) Assumption (2.21) is satisfied for the H 1/2 (Γ D ) orthogona projection and for the Scott Zhang projection for arbitrary p 1, asweasforthel 2 (Γ D ) projection for p =1. Current quasi optimaity resuts on AFEM rey on the fact that the estimator η 2 = ϱ2 +osc2 D, is equivaent to some inear convergent quasi-error quantity Δ. Whereas, the convergence theorem (Thm. 2.5) asohodsfor the usua Dörfer marking from [12], our contraction theorem reies on Stevenson s modification (2.11) (2.12). Moreover, we stress that the convergence theorem is constrained by the aprioriconvergence assumption (2.21), whereas the foowing contraction resut is not. Theorem 2.6 (contraction of AFEM). We use Agorithm 2.1 with (up to the genera assumptions stated above) arbitrary projection P and corresponding discrete soution U S 1 (T ) and estimator η. In addition, et P : H 1/2 (Γ D ) S p (E D) be the H1/2 (Γ D ) orthogona projection. Let Ũ S p (T ) be the Gaerkin soution of (2.5) with Ũ ΓD = P g and η 2 = ϱ 2 +osc2 D, be the associated error estimator from (2.9) with U repaced by Ũ. Then, for arbitrary 0 <θ 1,θ 2 < 1 and sufficienty sma 0 <ϑ<1, Agorithm 2.1 guarantees the existence of constants λ, μ > 0 and 0 <κ<1 such that the combined error quantity satisfies a contraction property Δ := (u Ũ) 2 L 2 (Ω) + λ g P g 2 H 1/2 (Γ D) + μ η 2 0 (2.23) Moreover, there are constants C ow,c high > 0 such that Δ +1 κδ for a N 0. (2.24) C ow Δ η 2 C high Δ. (2.25) In particuar, this impies convergence im u U H 1 (Ω) = 0 = im η of Agorithm 2.1 independenty of the precise choice of the uniformy H 1/2 (Γ D ) stabe projection P. Remark 2.7. The H 1/2 (Γ D ) orthogona projection is not needed for the impementation and can, in fact, hardy be computed expicity. Instead, it is ony used theoreticay for the numerica anaysis. More precisey, we wi see beow that the modified Dörfer marking (2.11) (2.12)fortheP chosen (with corresponding discrete soution U and error estimator η ) impies the usua Dörfer marking for the theoretica auxiiary probem with the H 1/2 (Γ D ) orthogona projection P and corresponding soution Ũ resp. error estimator η.

8 1214 M. AURADA ET AL. To state our quasi optimaity resut for Agorithm 2.1, we need to introduce further notation. Reca that, for a given trianguation T and M T, T +1 = refine(t, M ) (2.26) denotes the coarsest reguar trianguation such that a marked eements T M have been refined by (at east one) bisection. Moreover, we write T refine(t ) (2.27) if T is a finite refinement of T, i.e., there are finitey many trianguations T +1,...,T n and sets of marked eements M T,...,M n 1 T n 1 such that T = T n and T j+1 = refine(t j, M j ) for a j =,...,n 1. Finay, for a fixed initia mesh T 0,etT = { T : T refine(t 0 ) } be the set of a meshes which can be obtained by newest vertex bisection as we as the set T N = { T T :#T #T 0 N } of a trianguations which have at east N more eements than the initia mesh T 0. Reca that Agorithm 2.1 ony sees the error estimator η 2 = ϱ2 +osc2 D,, but not the error u U H 1 (Ω). From this point of view, it is natura to ask for the best possibe convergence rate for the error estimator. This can be characterized by means of an artificia approximation cass A s :Fors 0, we write (u, f, g, φ) A s def sup inf N s η <, (2.28) T T N where η 2 = ϱ 2 +osc 2 D, denotes the error estimator for the optima mesh T T N. By definition, this impies that a convergence rate η = O(N s ) is possibe if the optima meshes are chosen. The foowing theorem states that Agorithm 2.1, in fact, guarantees η = O(N s ) for the adaptivey generated meshes T. Theorem 2.8 (quasi optimaity of AFEM). Suppose that the sets M resp. M D in step (iii) (iv) of Agorithm 2.1 are chosen with minima cardinaity. Then, for sufficienty sma 0 < θ 1,ϑ < 1, but arbitrary 0 <θ 2 < 1, Agorithm 2.1 guarantees the existence of a constant C opt > 0 such that (u, f, g, φ) A s N η C opt (#T #T 0 ) s, (2.29) i.e. each possibe convergence rate s>0 is, in fact, asymptoticay obtained by AFEM. We stress that, up to now and as far as the error estimator is concerned, ony reiabiity (2.19) is needed for the anaysis. In particuar, the upper bounds on the sufficienty sma adaptivity parameters θ 1 and ϑ do not depend on the efficiency constant C eff from (2.20). This is in contrast to the preceding works on AFEM, e.g. [7, 10, 11, 14, 17, 27], which directy ask for optima convergence of the error (Thm. 2.9). Finay, the ower bound (2.20) for the error estimator aows to reate the approximation cass A s to the we-known definition from iterature in terms of the reguarity of the sought soution and the given data. In particuar, we obtain a quasi optimaity resut which is anaogous to those avaiabe in the iterature for homogeneous Dirichet data, see e.g. [5,10,27], but with ess dependencies for the upper bound of the adaptivity parameters. We refer to [6,15] for a characterization of approximation casses in terms of Besov reguarity. Theorem 2.9 (reation to usua approximation casses). It hods (u, f, g, φ) A s if and ony if the foowing four conditions hod: sup inf N N T T N N N min N s u V H 1 (Ω) <, (2.30) V S p (T ) sup inf N s osc T, <, N N T T N (2.31) sup inf N s osc N, <, N N T T N (2.32) sup inf N s osc D, <, T T N (2.33) N N i.e. the estimator and according to reiabiity hence the Gaerkin error converges with the best possibe rate aowed by the reguarity of the sought soution and the given data.

9 2.8. Outine AFEM WITH INHOMOGENEOUS DIRICHLET DATA 1215 Since our anaysis is strongy buit on properties of the Scott Zhang projection, Section 3 coects the essentia properties of the atter. This knowedge is used to prove Proposition 2.3. Moreover, we prove that the Scott Zhang error in a weighted H 1 (Γ D ) seminorm is (even ocay) equivaent to the Dirichet osciations (Prop. 3.1) which might be of genera interest. This aows to prove Proposition 2.4 with an estimator η which does not expicity contain the chosen projection P. Section 4 is concerned with the proof of Theorem 2.5. Section 5 gives the proof for the contraction resut of Theorem 2.6. Finay, the proofs of the quasi optimaity resuts of Theorems 2.8 and 2.9 are found in Section 6. Some numerica experiments in Section 7 concude the work. In a statements, the constants invoved and their dependencies are expicity stated. In proofs, however, we use the symbo to abbreviate up to a mutipicative constant which is independent of. Moreover, abbreviates that both estimates and hod. 3. Scott Zhang projection The main too of our anaysis is the Scott Zhang projection J : H 1 (Ω) S p (T ) (3.1) from [26]. A first appication wi be the proof of the Céa type estimate for the Gaerkin error (Prop. 2.3). Moreover, we prove that the Scott Zhang interpoation error in a ocay weighted H 1 seminorm is ocay equivaent to the Dirichet data osciations (Prop. 3.1). This wi be the main too to derive the bound (1 P )g H 1/2 (Γ D) osc D,, where the right-hand side is independent of the projection chosen Scott Zhang projection Anayzing the definition of J in [26], one sees that J can be defined ocay in the foowing sense: For an eement T T,thevaue(J w) T on T depends ony on the vaue of w ω,t on some eement patch { T ω,t T T : T } T. (3.2) For a boundary facet E E Γ, the trace of the Scott Zhang projection (J w) E on E depends ony on the trace w ω Γ,E on some facet patch { E ω,e Γ E E : E } E. (3.3) In case of a Dirichet facet E E D, one may choose ωγ,e Γ D. Moreover, J is defined in a way that the foowing projection properties hod: J W = W for a W S p (T ), (J w) Γ = w Γ for a w H 1 (Ω) andw S p (T )withw Γ = W Γ, (J w) ΓD = w ΓD for a w H 1 (Ω) andw S p (T )withw ΓD = W ΓD, i.e. the projection J preserves discrete (Dirichet) boundary data. Finay, J satisfies the foowing (oca) stabiity property (1 J )w L 2 (T ) C sz w L 2 (ω,t ) for a w H 1 (Ω) (3.4) and (oca) first order approximation property (1 J )w L 2 (T ) C sz h w L 2 (ω,t ) for a w H 1 (Ω) (3.5) where C sz > 0 depends ony on shape reguarity of T, cf. [26]. Here, h L (Ω) denotes the oca mesh width function defined by h T = T 1/d for a T T. Moreover, since the overap of the patches is controed in terms of shape reguarity, the integration domains in (3.4) (3.5) can be repaced by Ω, i.e. (3.4) (3.5) hodaso gobay.

10 1216 M. AURADA ET AL Scott Zhang projection onto discrete trace spaces We stress that J induces operators J Γ : L2 (Γ ) S p (E Γ ) and JD : L2 (Γ D ) S p (E D ) (3.6) in the sense of J Γ (w Γ )=(J w) Γ and J D (w Γ D )=(J Γ (w Γ )) ΓD for a w H 1 (Ω). We wi thus not distinguish these operators notationay. Arguing as in [26], for γ {Γ, Γ D,Γ N }, one sees that J satisfies even (oca) L 2 stabiity (oca) H 1 stabiity (1 J )w L 2 (E) C sz w L2 (ω Γ,E ) for a w L 2 (γ), (3.7) (1 J )w H 1 (E) C sz Γ w L 2 (ω Γ,E ) for a w H 1 (γ), (3.8) as we as a (oca) first order approximation property (1 J )w L2 (E) C sz h Γ w L 2 (ω Γ,E ) for a w H 1 (γ). (3.9) Here, Γ ( ) denotes again the surface gradient, and h L (Γ D ) denotes the oca mesh width function restricted to Γ D. According to shape reguarity of T, the integration domains in (3.7) (3.9) can be repaced by γ, i.e. (3.7) (3.9) hod aso gobay on γ. By standard interpoation arguments appied to (3.7) (3.8), one obtains stabiity (1 J )w H 1/2 (γ) C sz w H 1/2 (γ) for a w H 1/2 (γ) (3.10) in the trace norm. Moreover, it is proved in [18], Theorem 3 that the Scott Zhang projection satisfies (1 J )w H 1/2 (γ) C sz min W S p (T γ) h1/2 Γ (w W ) L2 (γ) for a w H 1 (γ). (3.11) Throughout, the constant C sz > 0 then depends ony on shape reguarity of T and on γ {Γ, Γ D,Γ N } Proof of Céa emma (Prop. 2.3) According to weak formuation (2.1) and Gaerkin formuation (2.5), we have the Gaerkin orthogonaity reation (u U ), V Ω =0 forav S p D (T ). Let L : H 1/2 (Γ ) H 1 (Ω) be a ifting operator. Let ĝ, ĝ H 1/2 (Γ ) denote arbitrary extensions of g = u ΓD resp. P g = U ΓD.Notethat(J LJ ĝ) ΓD =(J u) ΓD as we as (J LJ ĝ ) ΓD = U ΓD. For arbitrary V S p D (T ), we thus have U (V + J LJ ĝ ) S p D (T ), whence (u U ) 2 L 2 (Ω) = (u U ), (u (V + J LJ ĝ )) Ω according to the Gaerkin orthogonaity. Therefore, the Cauchy inequaity proves (u U ) L2 (Ω) min (u (V + J LJ ĝ )) V S p D (T L2 (Ω). ) We now pug in V = J u J LJ ĝ S p D (T ) and use stabiity of J and L to see (u U ) L 2 (Ω) (u J u + J LJ (ĝ ĝ )) L 2 (Ω) (u J u) L 2 (Ω) + ĝ ĝ H 1/2 (Γ ).

11 AFEM WITH INHOMOGENEOUS DIRICHLET DATA 1217 Since the extensions ĝ, ĝ of g and P g were arbitrary, we obtain (u U ) L 2 (Ω) (u J u) L 2 (Ω) + (1 P )g H 1/2 (Γ D). According to the projection property J W = W for W S p (T )andh 1 stabiity (3.4), it hods that (u J u) L2 (Ω) = The same argument for P with stabiity on H 1/2 (Γ )gives (1 P )g H 1/2 (Γ D) = Combining the ast three estimates, we infer (u U ) L 2 (Ω) + (1 P )g H 1/2 (Γ D) min (1 J )(u W ) L2 (Ω) min (u W ) L2 (Ω). W S p (T ) W S p (T ) min (1 P )(g W ΓD ) H W S p 1/2 (Γ D) min g W ΓD H (T ) W S p 1/2 (Γ (T ) D). ( ) min (u W ) L 2 (Ω) + g W ΓD H W S p (T ) 1/2 (Γ D). Finay, the Reich compactness theorem impies norm equivaence H 1 (Ω) ( ) L 2 (Ω) + ( ) ΓD H 1/2 (Γ D) on H 1 (Ω). This concudes the proof Scott Zhang projection and Dirichet data osciations We stress that the newest vertex bisection agorithm guarantees that ony finitey many shapes of eements T { T T : T T } can occur. In particuar, ony finitey many shapes of patches occur. Further detais are found in [30],Chapter4,asweasin[28,29]. This observation wi be used in the proof of the foowing emma. Proposition 3.1. Let Π : L 2 (Γ D ) P p 1 (E D ) denote the L2 (Γ D ) projection. Then, and, in particuar, (1 Π ) Γ g L2 (E) Γ (1 J )g L2 (E) C dir (1 Π ) Γ g L2 (ω Γ,E ) for a E E D (3.12) osc D, h 1/2 Γ (1 J )g L 2 (Γ D) C dir osc D, (3.13) The constant C dir 1 depends ony on Γ D, the poynomia degree p, the initia trianguation T 0,andtheuseof newest vertex bisection to obtain T T, butnot on g. Proof. Since Π is the piecewise L 2 projection, the ower bound in (3.12) (3.13) is obvious. To verify the upper bound, we argue by contradiction and assume that the upper bound in (3.12) is wrong for each constant C>0. For n N, we thus find some g n H 1 (Γ ) such that Γ (1 J ) g n L2 (E) >n (1 Π ) Γ g n L2 (ω Γ,E ). (3.14) Let Q,E : H 1 (ω,e Γ ) Sp (E ω Γ,E )denotetheh 1 orthogona projection on the patch ω,e Γ and define g n = (1 Q,E ) g n. Since the vaue of J v on E depends ony on the vaues of v on ω,e Γ, the projection property of J reveas (1 J )Q,E g n =0onE. Moreover, Γ Q,E g n P p 1 (E ω Γ,E )sothat(1 Π ) Γ Q,E g n =0onω,E Γ. From the orthogona decomposition g n = Q,E g n + g n,wethussee Γ (1 J )g n L 2 (E) = Γ (1 J ) g n L 2 (E) and (1 Π ) Γ g n L 2 (ω,e Γ ) = (1 Π ) Γ g n L 2 (ω,e Γ ). In particuar, we observe g n 0from(3.14) sothat we may define g n := g n / g n H1 (ω,e Γ ). This definition guarantees g n H 1 (ω Γ,E ) =1 and g n S p (E ω Γ,E ), (3.15)

12 1218 M. AURADA ET AL. where orthogonaity is understood with respect to the H 1 (ω,e Γ ) scaar product. Moreover, it hods that (1 Π ) Γ g n L2 (ω Γ,E ) < 1 n Γ (1 J )g n L2 (E) 1 n Γ g n L2 (ω Γ,E ) n 0 (3.16) due to the construction of g n and oca H 1 stabiity of J : H 1 (Γ D ) H 1 (Γ D ). First, (3.16) impies that Π Γ g n L2 (ω,e Γ ) C< is uniformy bounded as n.sinceπ Γ g n P p 1 (E ω Γ,E ) beongs to a finite dimensiona space, we may appy the Bozano Weierstrass theorem to extract a convergent subsequence. Without oss of generaity, we may thus assume Π Γ g n n Φ P p 1 (E ω Γ,E ) in strong L 2 sense. (3.17) Second, this and (3.16) provel 2 convergence of Γ g n to Φ, Γ g n Φ L2 (ω Γ,E ) (1 Π ) Γ g n L2 (ω Γ,E ) + Π Γ g n Φ L2 (ω Γ,E ) n 0. (3.18) Third, orthogonaity (3.15) impies g ω,e Γ n dγ = 0 if we consider the constant function 1 S p (E ω Γ,E ). Therefore, the Friedrichs inequaity and (3.18) predict uniform boundedness g n H 1 (ω,e Γ ) Γ g n L 2 (ω,e Γ ) C< as n. According to weak compactness in Hibert spaces, we may thus extract a weaky convergent subsequence. Without oss of generaity, we may thus assume g n n g H 1 (ω Γ,E ) in weak H1 sense. (3.19) Fourth, the combination of (3.18) and (3.19) impies Γ g = Φ. This foows from the fact that Φ Γ ( ) L2 (ω,e Γ ) is convex and continuous, whence weaky ower semicontinuous on H 1 (ω,e Γ ), i.e. Φ Γ g L 2 (ω,e Γ ) im inf n Φ Γ g n L 2 (ω,e Γ ) =0. Fifth, the Reich compactness theorem proves that the convergence in (3.19) does aso hod in strong L 2 sense. Together with (3.18) andφ = Γ g, we now observe strong H 1 convergence g g n 2 H 1 (ω Γ,E ) = g g n 2 L 2 (ω Γ,E ) + Φ Γ g n 2 L 2 (ω Γ,E ) n 0, whence g H1 (ω,e Γ ) = 1 as we as g S p (E ω Γ,E ) according to (3.15). On the other hand, Γ g = Φ P p 1 (E ω Γ,E ) impies g S p (E ω Γ,E ). This yieds g S p (E ω Γ,E ) S p (E ω Γ,E ) = {0} and contradicts g 2 H 1 (ω,e Γ ) =1. This contradiction proves the upper bound in (3.12). A standard scaing argument verifies that the constant C dir > 0 does ony depend on the shape of ω,e Γ but not on the diameter. As stated above, newest vertex bisection guarantees that ony finitey many shapes of patches ω,e Γ may occur, i.e. C dir > 0 depends ony on T 0 and the use of newest vertex bisection. Summing (3.12) over a Dirichet facets, we see osc 2 D, = h1/2 (1 Π ) Γ g 2 L 2 (Γ D) h1/2 Γ (1 J )g 2 L 2 (Γ D) h E (1 Π ) Γ g 2 L 2 (ω,e Γ ) h 1/2 (1 Π ) Γ g 2 L 2 (Γ, D) where the fina estimate hods due to uniform shape reguarity. Coroary 3.2. It hods (1 P )g H 1/2 (Γ D) C osc osc D,,whereC osc > 0 depends on Γ D, the poynomia degree p 1, stabiity C stab > 0, the initia mesh T 0, and the use of newest vertex bisection. Proof. By use of the projection property and stabiity of P, one sees (1 P )g H 1/2 (Γ D) = (1 P )(1 J )g H 1/2 (Γ D) (1 J )g H 1/2 (Γ D). The approximation estimate (3.11) and Proposition 3.1 concude (1 J )g H 1/2 (Γ D) h 1/2 Γ (1 J )g L 2 (Γ D) osc D,. E E D

13 3.5. Proof of reiabiity and efficiency (Prop. 2.4) We consider a continuous auxiiary probem AFEM WITH INHOMOGENEOUS DIRICHLET DATA 1219 Δw =0 inω, w =(1 P )g on Γ D, n w =0 onγ N, (3.20) with unique soution w H 1 (Ω). We then have norm equivaence w H 1 (Ω) (1 P )g H 1/2 (Γ D) as we as u U w HD 1 (Ω). From this, we obtain u U H1 (Ω) (u U w) L2 (Ω) + (1 P )g H 1/2 (Γ D). The first term on the right hand side can be handed as for homogeneous Dirichet data, i.e. use of the Gaerkin orthogonaity combined with approximation estimates for a Cément type quasi interpoation operator (e.g. the Scott Zhang projection). This eads to (u U w) L2 (Ω) ϱ Detais are found e.g. in [4]. The H 1/2 (Γ D ) norm is dominated by the Dirichet data osciations osc D,,see Coroary 3.2. By use of bubbe functions and oca scaing arguments, one obtains the estimates T 2/d f + ΔU 2 L 2 (T ) (u U ) 2 L 2 (T ) +osc T,(T ) 2 +osc N, ( T Γ N ), T 1/d [ n U ] 2 L 2 (E Ω) (u U ) 2 L 2 (Ω,E ) +osc T,(ω,E ) 2, T 1/d φ n U 2 L 2 (E Γ N ) (u U ) 2 L 2 (Ω,E ) +osc T,(ω,E ) 2 +osc N, (E Γ N ) 2, where Ω,E = T + T denotes the facet patch of T + T = E E. Detais are found e.g. in [3,30]. Summing these estimates over a eements, one obtains the efficiency estimate (2.20). 4. Convergence In this section, we aim to prove Theorem 2.5. Our proof of the convergence theorem reies on the estimator reduction principe from [2], i.e. we verify that the error estimator is contractive up to some zero sequence Estimator reduction estimate Note that the estimator η 2 = ϱ2 +osc2 D, can be ocaized over eements via η 2 = T T η (T ) 2 with η (T ) 2 = ϱ (T ) 2 + T 1/d (1 Π ) Γ g 2 L 2 ( T Γ D) (4.1) with Π : L 2 (Γ D ) P p 1 (E D )the(evened piecewise) L2 (Γ D ) orthogona projection. Lemma 4.1 (modified marking impies Dörfer marking). For arbitrary 0 <θ 1,θ 2,ϑ < 1 in Agorithm 2.1, there is some parameter 0 <θ<1 such that the error estimator η 2 = ϱ2 +osc2 D, satisfies θη 2 and a eements T M are refined by at east one bisection. T M η (T ) 2, (4.2)

14 1220 M. AURADA ET AL. Proof. First, assume osc 2 D, ϑϱ2 and et M T satisfy (2.11). Then, θ 1 (ϱ 2 +osc 2 D,) θ 1 (1 + ϑ)ϱ 2 (1 + ϑ) T M ϱ (T ) 2. Therefore, the Dörfer marking (4.2) hodswithθ θ 1 (1 + ϑ) 1. Second, assume osc 2 D, >ϑϱ2 and et MD ED satisfy (2.12). Then, θ 2 (ϱ 2 +osc 2 D,) θ 2 (1 + ϑ 1 )osc 2 D, (1 + ϑ 1 ) osc D, (E) 2. Therefore, the Dörfer marking (4.2) hodswithθ θ 2 (1 + ϑ 1 ) 1, and a eements which have some facet E M D are refined. Proposition 4.2 (estimator reduction). Let T refine(t ) be an arbitrary refinement of T and M T \T a subset of the refined eements which satisfies the Dörfer marking (4.2) for some 0 <θ<1. Then, E M D η 2 q red η 2 + C red (U U ) 2 L 2 (Ω) (4.3) with certain constants 0 <q red < 1 and C red > 0 which depend ony on the parameter 0 <θ<1, shape reguarity of T, and the poynomia degree p 1. Proof. The proof foows aong the ines of the proof for homogeneous Dirichet data, which is found in [10], Coroary 3.4, Proof of Theorem 4.1. For detais, the reader is referred to the extended preprint of this work [1] Aprioriconvergence of Scott Zhang projection We assume that (J +1 v) T =(J v) T for a T T T +1 with ω,t (T T +1 )whichcanawaysbe achieved by an appropriate choice of the dua basis functions in the definition of J +1. In this section, we prove that under the aforegoing assumptions and for arbitrary refinement, i.e. T = refine(t 1 ) for a N, the imit of the Scott Zhang interpoants J v exists in H 1 (Ω) as. In particuar, this provides the essentia ingredient to prove that, under the same assumptions, the imit of Gaerkin soutions U exists in H 1 (Ω). For 2D and first order eements p = 1, this resut has first been proved in [14], Lemma 18. The proof transfers directy to the present setting without any other but the obvious modifications. Proposition 4.3. Let v H 1 (Ω). Then, the imit J v := im J v exists in H 1 (Ω) and defines a continuous inear operator J : H 1 (Ω) H 1 (Ω). Coroary 4.4. Under the assumptions of Proposition 4.3, theimitg := im J g exists in H 1/2 (Γ D ). Proof. Let ĝ H 1/2 (Γ ) denote an arbitrary extension of g. With some ifting operator L, we define v := Lĝ and note that (J v) ΓD =(J ĝ) ΓD = J g.sincej v = im J v exists in H 1 (Ω), we obtain (J v) ΓD J g H 1/2 (Γ D) (J v) Γ J ĝ H 1/2 (Γ ) J v J v H1 (Ω) 0. This concudes the proof with (J v) ΓD =: g Aprioriconvergence of orthogona projections In this subsection, we reca an eary observation from [8], Lemma 6.1 which wi be appied severa times. We stress that the origina proof of [8] is based on the orthogona projection. However, the argument aso works for (possiby noninear) projections with P P k = P for k which satisfy a Céa type quasi optimaity. Since the Scott Zhang projection satisfies J J k J, in genera, Proposition 4.3 is not a consequence of such an abstract resut.

15 AFEM WITH INHOMOGENEOUS DIRICHLET DATA 1221 Lemma 4.5. Let H be a Hibert space and X be a sequence of cosed subspaces with X X +1 for a 0. Let P : H X denote the H orthogona projection onto X. Then, for each x H, theimitx := im P x exists in H. Since the discrete trace spaces S p (E D ) are finite dimensiona and hence cosed subspaces of H1/2 (Γ D ), the emma immediatey appies to the H 1/2 (Γ D ) orthogona projection. Coroary 4.6. Let P : H 1/2 (Γ D ) S p (E D) denote the H1/2 (Γ D ) orthogona projection. Then, the imit g := im P g exists in H 1/2 (Γ D ). Coroary 4.7. Let π : L 2 (Γ D ) S 1 (E D) denote the L2 (Γ D ) orthogona projection. Then, the imit g := im π g exists weaky in H 1 (Γ D ) and strongy in H α (Γ D ) for a 0 α<1. Proof. According to Lemma 4.5, the imit g = im π g exists strongy in L 2 (Γ D ). Moreover and according to [19], Theorem 6, the π are uniformy stabe in H 1 (Γ D ), since we use newest vertex bisection. Hence, the sequence (π g) is uniformy bounded in H 1 (Γ D ) and thus admits a weaky convergent subsequence (π k g)with weak imit g H 1 (Γ D ), where weak convergence is understood in H 1 (Γ D ). Since the incusion H 1 (Γ D ) L 2 (Γ D ) is compact, the sequence (π k g) converges strongy to g in L 2 (Γ D ). From uniqueness of imits, we concude g = g. Iterating this argument, we see that each subsequence of (π g) contains a subsequence which converges weaky to g in H 1 (Γ D ). This proves that the entire sequence converges weaky to g in H 1 (Γ D ). Strong convergence in H α (Γ D ) foows by compact incusion H 1 (Γ D ) H α (Γ D ) for a 0 α< Aprioriconvergence of Gaerkin soutions We now show that the imit of Gaerkin soutions U exists as provided that the meshes are nested, i.e. T +1 = refine(t ). Proposition 4.8. Under Assumption (2.21) that g := im P g exists in H 1/2 (Γ ), aso the imit U := im U of Gaerkin soutions exists in H 1 (Ω). Proof. We consider the continuous auxiiary probem Δw =0 inω, w = P g on Γ D, n w =0 onγ N. Let w H 1 (Ω) be the unique (weak) soution and note that the trace ĝ := w Γ extension of P g with H 1/2 (Γ )providesan ĝ H 1/2 (Γ ) w H 1 (Ω) P g H 1/2 (Γ D) ĝ H 1/2 (Γ ). For arbitrary k, N, the same type of arguments proves ĝ ĝ k H 1/2 (Γ ) (P P k )g H 1/2 (Γ D). According to Assumption (2.21), (P g) is a Cauchy sequence in H 1/2 (Γ D ). Therefore, ĝ is a Cauchy sequence in H 1/2 (Γ ), whence convergent with imit ĝ H 1/2 (Γ ). Next, note that (J Lĝ ) ΓD = P g,wherel : H 1/2 (Γ ) H 1 (Ω) denotes some ifting operator. Therefore, Ũ := U J Lĝ S p D (T ) is the unique soution of the variationa formuation Ũ, V Ω = u, V Ω J Lĝ, V Ω for a V S p D (T ). (4.4) Finay, we need to show that Ũ and J Lĝ are convergent to concude convergence of U = Ũ + J Lĝ.

16 1222 M. AURADA ET AL. With convergence of (ĝ )toĝ and Proposition 4.3, weobtain J Lĝ J Lĝ H 1 (Ω) J (Lĝ Lĝ ) H 1 (Ω) + J Lĝ J Lĝ H 1 (Ω) ĝ ĝ H 1/2 (Γ ) + J Lĝ J Lĝ H1 (Ω) 0. This proves convergence of J Lĝ to J Lĝ as. To see convergence of Ũ, etũ, S p D (T )bethe unique soution of the discrete auxiiary probem Ũ,, V Ω = u, V Ω J Lĝ, V Ω for a V S p D (T ). (4.5) Due to the nestedness of the ansatz spaces S p D (T ), Lemma 4.5 predicts aprioriconvergence Ũ, Ũ H 1 (Ω D ). With the stabiity of (4.4) and(4.5), we obtain and therefore Ũ Ũ in H 1 (Ω D ). Finay, we now deduce (Ũ, Ũ) L 2 (Ω) J Lĝ J Lĝ H 1 (Ω) U = Ũ + J Lĝ 0, Ũ + J Lĝ =: U H 1 (Ω), which concudes the proof Proof of convergence theorem (Thm. 2.5) (i) Since the imit U = im U exists in H 1 (Ω), we infer im (U +1 U ) L 2 (Ω) = 0. In view of this and Lemma 4.1, the estimator reduction estimate (4.3) takes the form η 2 +1 q red η 2 + α for a 0 with some non negative α 0 such that im α =0,i.e. the estimator is contractive up to a non negative zero sequence. It is a consequence of eementary cacuus that im η =0,seee.g. [2], Lemma 2.3. Finay, reiabiity u U H 1 (Ω) η thus concudes the proof. (ii) The verification of Assumption (2.21) is done in Coroary 4.4 for the Scott Zhang projection, Coroary 4.6 for the H 1/2 (Γ D ) orthogona projection, and Coroary 4.7 for the L 2 (Γ D ) orthogona projection. 5. Contraction In principe, the convergence rate of im U = u from Theorem 2.5 coud be sow. Moreover, Theorem 2.5 restricts the Dirichet projection P by Assumption (2.21). In this section, we aim to show inear convergence for some quasi-error quantity Δ η 2 = ϱ2 +osc2 D, with respect to the step of Agorithm 2.1 and independenty of the projection P chosen. The essentia observation is that the marking step in Agorithm 2.1 is in some sense independent of the P chosen Impicit Dörfer marking Let Ũ S p (T ) be a Gaerkin soution of (2.5) with different Dirichet data Ũ = P g on Γ D,where P : H 1/2 (Γ D ) S p (E D) is a uniformy stabe projection onto Sp (E D 2 ) in the sense of (2.17). Let η = ϱ 2+osc2 D, be the associated error estimator. In the foowing, we prove that marking in Agorithm 2.1 with η 2 = ϱ2 +osc2 D, and sufficienty sma 0 <ϑ<1 impicity impies the simpe Dörfer marking (4.2) for η.

17 AFEM WITH INHOMOGENEOUS DIRICHLET DATA 1223 Lemma 5.1 (oca equivaence of error estimators for different projections). For arbitrary U T, it hods that C 1 eq T U ϱ (T ) 2 ϱ (T ) 2 +osc 2 D, and Ceq 1 T U T U ϱ (T ) 2 T U ϱ (T ) 2 +osc 2 D,. (5.1) The constant C eq > 1 depends ony on shape reguarity of T and on C stab > 0. In particuar, this impies equivaence (C eq +1) 1 η 2 η 2 (C eq +1)η 2. (5.2) Proof. Arguing as for the estimator reduction, it foows from the triange inequaity and scaing arguments that ϱ (T ) 2 ϱ (T ) 2 + (U Ũ) 2 L 2 (ω (T )) for a T T, where ω (T )= { T T : T T } denotes the eement patch of T. Consequenty, a rough estimate gives ϱ (T ) 2 ϱ (T ) 2 + (U Ũ) 2 L 2 (Ω) for a U T. T U T U Reca the Gaerkin orthogonaity (U Ũ), V = (u Ũ), V (u U ), V =0 forav S p D (T ). Let ĝ H 1/2 (Γ ) be an arbitrary extension of (U Ũ) ΓD function V =(U Ũ) J Lĝ SD 1 (T )tosee =(P P )g H 1/2 (Γ D ). We choose the test (U Ũ) 2 L 2 (Ω) = (U Ũ), J Lĝ Ω. Stabiity of Scott Zhang projection J and ifting operator L thus give (U Ũ) L 2 (Ω) J Lĝ L 2 (Ω) ĝ H 1/2 (Γ ). Since ĝ was an arbitrary extension of (P P )g, we end up with (U Ũ) L 2 (Ω) (P P )g H 1/2 (Γ D) (P 1)g H 1/2 (Γ D) + (1 P )g H 1/2 (Γ D) osc D,, where we have used Coroary 3.2. This proves the first estimate in (5.1), and the second foows with the same arguments. The foowing emma is the main reason, why we stick with Stevenson s modified Dörfer marking (2.11) (2.12) instead of simpe Dörfer marking (4.2). Lemma 5.2 (modified Dörfer marking impies Dörfer marking for different projection). For arbitrary 0 < θ 1,θ 2 < 1 and sufficienty sma 0 <ϑ<1, thereissome0 <θ<1 such that the marking criterion (2.11) (2.12) for η 2 = ϱ2 +osc2 D, impies the Dörfer marking θ η 2 η (T ) 2 (5.3) T M for η 2 = ϱ 2 +osc2 D,.Theparameter0 <θ<1 depends on 0 <θ 1,θ 2,ϑ<1 and on C eq > 0.

18 1224 M. AURADA ET AL. Proof. We argue as in the Proof of Lemma 4.1. First, assume osc 2 D, ϑϱ2 and et M T satisfy (2.11). AccordingtoLemma5.1, wesee ( ) θ 1 η 2 θ 1(1 + ϑ)ϱ 2 (1 + ϑ) ϱ (T ) 2 C eq (1 + ϑ) ϱ (T ) 2 +osc 2 D, T M T M ( ) C eq (1 + ϑ) ϱ (T ) 2 + ϑϱ 2. T M This proves ( θ1 C 1 eq (1 + ϑ) 1 ϑ ) η 2 T M ϱ (T ) 2. Together with (C eq +1) 1 η 2 η2, we thus obtain the Dörfer marking (5.3)with0<θ (C eq+1) 1( θ 1 C 1 eq (1+ ϑ) 1 ϑ ) < 1, provided that 0 <ϑ<1 is sufficienty sma compared to 0 <θ 1 < 1. Second, assume osc 2 D, >ϑϱ2 and et MD ED satisfy (2.12). Then, θ 2 η 2 θ 2(1 + ϑ 1 )osc 2 D, (1 + ϑ 1 ) E M D osc D, (E) 2 (1 + ϑ 1 ) T M η (T ) 2, where M = { T T : E M D E T } is defined in step (iv) of Agorithm 2.1.Asbefore(C eq +1) 1 η 2 thus proves (4.2) with0<θ C 1 eq θ 2 (1 + ϑ 1 ) 1 < 1. η Quasi Pythagoras theorem To prove Theorem 2.6, weconsideratheoretica auxiiary probem: Throughout the remainder of Section 5, Ũ S p (T ) denotes the Gaerkin soution of (2.5) with Dirichet data Ũ = P g on Γ D,whereP : H 1/2 (Γ D ) S p (E D) denotes the H1/2 (Γ D ) orthogona projection. Associated with Ũ is the error estimator η 2 = ϱ 2+osc2 D,, where ϱ is defined in (2.10) withu repaced by Ũ. Reca that the aforegoing statements of Section 3 and Section 4 hod for any uniformy H 1/2 (Γ D ) stabe projection P and thus appy to η 2 = ϱ 2 +osc2 D,. We sha need reiabiity u Ũ 2 2 H 1 (Ω) η as we as the estimator reduction (4.3) from Proposition 4.2 for η 2, which is a consequence of Lemma 5.2. Our concept of proof of Theorem 2.6 goes back to [10], proof of Theorem 4.1. Therein, however, the proof reies on the Pythagoras theorem (u U ) 2 L 2 (Ω) = (u U +1) 2 L 2 (Ω) + (U +1 U ) 2 L 2 (Ω) which does not hod in case of inhomogeneous Dirichet data and P g P +1 g, in genera. Instead, we rey on a quasi Pythagoras theorem which wi be used for the auxiiary probem. Lemma 5.3 (quasi Pythagoras theorem). Let T refine(t ) be an arbitrary refinement of T with the associated auxiiary soution Ũ S p (T ),whereũ = P g on Γ D.Then, (1 α) (u Ũ ) 2 L 2 (Ω) (u Ũ) 2 L 2 (Ω) (Ũ Ũ) 2 L 2 (Ω) + α 1 C pyth (P P )g 2 H 1/2 (Γ D) (5.4) for a α>0. The constant C pyth > 0 depends ony on the shape reguarity of σ(t ) and σ(t ) and on Ω and Γ D. Proof. For p = 1 and noda interpoation for 2D, a simiar resut is found in [22], Lemma 5.1 or [14], Lemma 12. Essentiay the same arguments can aso be empoyed here. Detais are found in the extended preprint of this work [1].

19 5.3. Proof of contraction theorem (Thm. 2.6) AFEM WITH INHOMOGENEOUS DIRICHLET DATA 1225 Using the quasi Pythagoras theorem (5.4) witht = T +1,wesee (1 α) (u Ũ+1) 2 L 2 (Ω) (u Ũ) 2 L 2 (Ω) (Ũ+1 Ũ) 2 L 2 (Ω) + α 1 C pyth (P +1 P )g 2 H 1/2 (Γ D). The use of the H 1/2 (Γ D ) orthogona projection provides the orthogonaity reation (1 P +1 )g 2 H 1/2 (Γ D) + (P +1 P )g 2 H 1/2 (Γ D) = (1 P )g 2 H 1/2 (Γ D). Combining the ast two estimates, we obtain (1 α) (u Ũ+1) 2 L 2 (Ω) + α 1 C pyth (1 P +1 )g 2 H 1/2 (Γ D) (u Ũ) 2 L 2 (Ω) + α 1 C pyth (1 P )g 2 H 1/2 (Γ D) (Ũ+1 Ũ) 2 L 2 (Ω). Appying Lemma 5.2, we see that Agorithm 2.1 for η 2 = ϱ2 +osc2 D, impicity impies the Dörfer marking (5.3) (resp. (4.2)) for η 2 = ϱ 2 +osc2 D,. Therefore, the estimator reduction (4.3) ofproposition4.2 appies to the auxiiary probem and provides η +1 2 q red η 2 + C red (Ũ+1 Ũ) 2 L 2 (Ω) for a 0. Now, we add the ast two estimates to see, for β>0, (1 α) (u Ũ+1) 2 L 2 (Ω) + α 1 C pyth (1 P +1 )g 2 H 1/2 (Γ D) + β η 2 +1 (u Ũ) 2 L 2 (Ω) + α 1 C pyth (1 P )g 2 H 1/2 (Γ D) + βq red η 2 +(βc red 1) (Ũ+1 Ũ) 2 L 2 (Ω). We choose β>0sufficienty sma to guarantee βc red 1 0, i.e. the ast term on the right hand side of the ast estimate can be omitted. Then, we use the reiabiity u Ũ 2 2 H 1 (Ω) η and the estimate (1 P )g 2 H 1/2 (Γ D) osc 2 D, η 2 from Coroary 3.2 in the form to see, for arbitrary γ,δ > 0 (u Ũ) 2 L 2 (Ω) + (1 P )g 2 H 1/2 (Γ D) C η 2 (1 α) (u Ũ+1) 2 L 2 (Ω) + α 1 C pyth (1 P +1 )g 2 H 1/2 (Γ D) + β η 2 +1 (u Ũ) 2 L 2 (Ω) + α 1 C pyth (1 P )g 2 H 1/2 (Γ D) + βq red η 2 (1 γβc 1 ) (u Ũ) 2 L 2 (Ω) +(1 δβc 1 )α 1 C pyth (1 P )g 2 H 1/2 (Γ D) + β(q red + γ + δα 1 C pyth ) η 2. For 0 <α<1, we may now rearrange this estimate to end up with (u Ũ+1) 2 L 2 (Ω) + 1 γβc 1 1 α C pyth α(1 α) (1 P +1)g 2 H 1/2 (Γ + β D) 1 α η +1 2 (u Ũ) 2 L 2 (Ω) +(1 δβc 1 ) +(q red + γ + δα 1 C pyth ) β 1 α η 2. C pyth α(1 α) (1 P )g 2 H 1/2 (Γ D)