ON THE STRONG CONVERGENCE OF GRADIENTS IN STABILIZED DEGENERATE CONVEX MINIMIZATION PROBLEMS

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1 SIAM J NUMER ANAL Vo 47, No 6, pp c 010 Society for Industria and Appied Mathematics ON THE STRONG CONVERGENCE O GRADIENTS IN STABILIZED DEGENERATE CONVEX MINIMIZATION PROBLEMS WOLGANG BOIGER AND CARSTEN CARSTENSEN Abstract Infimizing sequences in nonconvex variationa probems typicay exhibit enforced finer and finer osciations caed microstructures such that the infima energy is not attained Athough those osciations are physicay meaningfu, finite eement approximations experience difficuty in their reconstruction The reaxation of the nonconvex minimization probem by semi convexification eads to a macroscopic mode for the effective energy The resuting discrete macroscopic probem is degenerate in the sense that it is convex but not stricty convex This paper studies a modified discretization by adding a stabiization term to the discrete energy It wi be proven that for a wide cass of probems, this stabiization technique eads to strong H 1 convergence of the macroscopic variabes even on unstructured trianguations In contrast to the work [C Carstensen, P Pecháĉ, S Bartes, and A Proh, Interfaces ree Bound, 6 004, pp 53 69] on quasi-uniform trianguations, this paper aows for genera unstructured shape-reguar trianguations and so enabes the use of adaptive agorithms for the stabiized formuations Key words adaptive finite eement methods, cacuus of variations, convexification, degenerate convex probems, energy reduction, nonconvex minimization, partia differentia equation, reaxation, stabiization, strong convergence, variationa probem AMS subject cassifications 35J70, 65K10, 65N1, 65N30 DOI / Introduction It is we known that variationa probems with non quasi- convex energy density have, in genera, no cassica soution [8], [5], [], [6], [9] Minimizing sequences deveop enforced finer and finer osciations with a Young-measure vaued imit Moreover, those osciations make it hard for numerica methods to sove reeated discrete probems Reaxation methods provide macroscopic modes, which repace the nonconvex energy density by a semi convex enveope A range of exampes and appications incudes the noninear Lapacian, some optima design probems, severa convexified mode probems in computationa microstructure ike the three-we probem, and eastopasticity, discussed in [5] and [] The ack of strict convexity yieds situations where the Hessian matrix is not definite, and so minimization agorithms encounter severe difficuties eg, the unstabiized three-we probem does not aow a Newton sove A straightforward remedy is a stabiization technique, where the probem is reguarized by adding a positive semidefinite stabiization term to the discrete energy Such methods and especiay different choices for the stabiization have been proposed in [3], where it is proven that proper stabiization can even yied strong H 1 convergence when working with quasi-uniform trianguations Received by the editors January 13, 009; accepted for pubication in revised form October 13, 009; pubished eectronicay ebruary 17, 010 This work was supported by DG Research Center MATHEON in Berin Department of Mathematics, Humbodt Universität zu Berin, Unter den Linden 6, Berin, Berin, Germany, and Department of Computationa Science and Engineering, Yonsei University, Seou, Korea boiger@mathhu-berinde, cc@mathhu-berinde The first author s research was supported by the Deutsche orschungsgemeinschaft DG through the RTG 118 as we as by the Berin Mathematica Schoo BMS The second author s research was supported by the Deutsche orschungsgemeinschaft DG through the Research Group 797 Micropast 4569 Copyright by SIAM Unauthorized reproduction of this artice is prohibited

2 4570 WOLGANG BOIGER AND CARSTEN CARSTENSEN This paper suggests a modified stabiization of [3, equation 61] which yieds H 1 convergence even on possiby nonquasi-uniform trianguations and hence enabes adaptive mesh-refinement Strong convergence is a promising property, for it enabes the computation of quantities reated to the soution or exampe, voume fractions of Young-measure vaued soutions of the origina nonconvex probem are nonineary dependent of the soution of the convexified probem and thus cannot be derived from weaky converging approximations Based on this resut, the next chaenge is finding reiabe and efficient error estimators in order to contro the adaptive grid refinement The mode probem anayzed is defined as foows: Given a bounded Lipschitz domain R n n =, 3 with poygona boundary, a fixed p, and m N, et u D W 1,p ; R m C; R m be piecewise W,p in and piecewise H on the boundary see beow or section 1 for detais The set of admissibe functions reads A := V + u D for V = W 1,p 0 ; R m With a continuous convex energy density W : R m n R and some ower-order term A R,v Lx, vx dx with a continuous convex L : Rm R, the convex minimization probem reads 11 minimize Ev := W vx dx + Lx, vx dx amongst v A A finite eement approximation of 11 is associated with a famiy of reguar trianguations T N0 of in the sense of Ciaret [7] eg, for n =, the intersection of two trianges is either a common node or a common edge or empty, the set of inner edges or faces of T,andH := max T T diamt Let u D, S 1 T ; R m be the noda interpoation of u D,whereS 1 T ; R m contains the owest-order conforming R m -vaued finite eement functions on T,and A := V + u D, for V = V P 1 T, R m or ater error estimates, we assume u D to be T 0 -piecewise W,p and 0 - piecewise H or functions on which are discontinuous on the edges or faces of the trianguations, the jump of such a function v on a face or edge shared by two trianges or tetrahedra T + and T reads [v]x =[v] x = im vy T + y x The stabiization reads a v, w := im T y x vy for x H [ v] :[ w]ds, diam where : denotes the scaar product in R m n The discrete probem reads 1 minimize E v :=Ev+ 1 a v, v amongst v A We denote SX :=DW X forx R m n, Jv; w = DLx, vx; wx dx for v, w W 1,p ; R m Copyright by SIAM Unauthorized reproduction of this artice is prohibited

3 STRONG CONVERGENCE IN CONVEX MINIMIZATION 4571 where DL is the derivative of L with respect to the second argument, and J v; w :=Jv; w+a v, w The Euer Lagrange equations of 11 1 consist in finding u Aand u A with 13 S u : v dx + Ju; v = 0 for a v V, 14 S u : v dx + J u ; v = 0 for a v V or the convexity conditions of section, the main resut of this paper is Theorem 44, which states that a unique continuous soution u A W,p T 0 H 3/+ε with ε>0 satisfies 15 u u H 1 H The remainder of this paper is devoted to the proof of 15 and is organized as foows: Section presents the genera framework and ists a the reguarity that is demanded of the continuous probem Section 3 discusses the genera error estimate of [3, Theorem 1] inay, section 4 presents the proof of the main resut Here and throughout this paper, we empoy standard notation on Lebesgue and Soboev spaces, and we wi often abbreviate spaces of vector- or matrix-vaued functions, whenever the dimensionaity is cear, eg, v W 1,p = v L p = v L p ;R m n for v W 1,p = W 1,p ; R m urthermore, W,p T ; R m := { v : R : v T W,p T ; R m for a T T } and H T ; R m =W, T ; R m In the foowing, a b abbreviates a Cb with a constant C>0 independent of, and a b abbreviates a b a Prerequisites and assumptions In this section we wi state some assumptions on the given trianguations and functions which wi be needed in the foowing proofs 1 Trianguations Let T N0 be a famiy of reguar trianguations of in the sense of Ciaret [7] or each trianguation T,wedenotethesetofitsedgesor faces with,asweasitssubsets = { : } and = \ or a given, we denote the patch of with ω = T T T We further define T h T := diamt andh := diam fort T and,soh =max T T h T We assume the famiy T N0 to be shape-reguar in the sense that h T h for a T T, with T The initia trianguation T 0 is assumed to be fine enough, and it is aso assumed that u D W,p T 0 satisfies u D H foreach 0 The noda interpoation operator I : C S 1 T ons 1 T is defined by I vz =vz for a v C and z N eg, u D, = I u D Copyright by SIAM Unauthorized reproduction of this artice is prohibited

4 457 WOLGANG BOIGER AND CARSTEN CARSTENSEN The trace inequaity a consequence of [1, Theorem 166] reads as foows: or any v H 1 and T T with T, v L v L T v L T + h 1 v L T h v L T + h 1 v L T The foowing estimates can be found in [1, Theorem 444] Let T be a triange or tetrahedron with diameter h T =diamt, and for t>1, et v W,t T with v = 0 on each node of T ;then 1 v L t T + h T v L t T h T v Lt T Assumptions on the energy density This subsection presents the assumptions on the energy density W The foowing assumption is simiar to [3, H1] with the robenius matrix norm induced by the scaar product : in R m n Assumption 1convexity contro There are α, r, s > 0with1<r and s + r + p rp such that, for a X, Y R m n, it hods that SX SY r α 1 + X s + Y s SX SY : X Y Since S = DW is the derivative of the energy density function W : R m n R, condition foows from SX SY r α 1 + X s + Y s W X W Y SY :X Y Assumption growth conditions X p 1 W X 1+ X p for a X R m n There is a fixed q with 0 <q<psuch that the ower-order term satisfies Lx, vx dx 1+ v q W 1,p for a v A The partia derivatives ξj Lx, ξ of the integrand are Carathéodory functions in the sense of [8, Definition 35] ie, ξj Lx, is continuous for amost every x, and ξj L,ξ is measurabe for every ξ R m, the Jacobian DLx, ξ; R m with respect to the second argument of L satisfies for every ξ R m, and amost every x DLx, ξ; ζx+ ξ p 1, with an ζ L p, where p is the conjugate exponent of p, 1/p +1/p =1 Coroary 1 The set of minimizers of the continuous and discrete probems 11 and 1 are the set of soutions of the corresponding Euer Lagrange equations 13 and 14 Proof This is a consequence of the ast paragraph in Assumption and [8, Theorem 337] Copyright by SIAM Unauthorized reproduction of this artice is prohibited

5 STRONG CONVERGENCE IN CONVEX MINIMIZATION Stabiization via jumps of gradients As described in the introduction, we impement a stabiization function a : H T ; R m H T ; R m R In this paper, we wi study a cass of stabiization functions which penaize jumps of gradients and which are defined by 3 a v, w := ρ [ v] :[ w]ds, where ρ := H 1+γ /h for,and 1 <γ<3 These stabiization functions are modifications of [3, equation 61] With this, we define J := J + a on H T ; R m H T ; R m urthermore, we define a seminorm by = a, Here and throughout the paper we assume there exist u A W,p T 0 ; R m H 3/+ε ; R m withε>0 and u A such that σ := S u andσ := S u satisfy 13 14, which reads σ : v dx + Ju; v = 0 for a v V, 4 σ : v dx + J u ; v = 0 for a v V We denote e := u u and δ := σ σ 4 Assumptions on ow-order terms The foowing assumptions on the derivative J of the ow-order term are simiar to [3, H4 H5], where u, σ and u,σ are defined as in the preceding subsection Ony one of these aternative assumptions needs to be satisfied for the main theorem to be appicabe Assumption 3 There exist 0 <m M< such that β := m e L Ju; e Ju ; e, Ju; v Ju ; v M e L v L for a v W 1,p ; R m urthermore, we define ζ := r/r 1 Assumption 4 There exist M > 0, a constant vector z R, z =1,anda constant C z > 0 such that 5 β := 0 Ju; e Ju ; e, Ju; v Ju ; v M e L v L for a v W 1,p ; R m, z e L C z δ : e dx urthermore, we define ζ := 5 Exampe: Scaar two-we probem Let X 1,X R with X 1 X and define the energy density W : R R, X X X 1 X X The space of admissibe functions is here given by A := W 1,4 0 + u D with a fixed function u D W 1,4 with u D H Let W be the convex hu of W ; then the convexified energy function E : A R is Ev = W vdx + λ f v L gv dx, Copyright by SIAM Unauthorized reproduction of this artice is prohibited

6 4574 WOLGANG BOIGER AND CARSTEN CARSTENSEN where f,g L are fixed functions and λ 0 is some fixed constant Here the derivative J of the ower-order term reads as foows: Jv; w = λv f g w dx The minimization of E in A can be discretized according to 1, with the stabiization function 3 for γ = 1; thatis a v, w := H [ v] :[ w]ds h According to [4, Coroary 1], for any X, Y R and Y 1 =X X 1 / and Y =X + X 1 /, SY SX 16 max {1, Y 1 + Y } 1+ X + Y SY SX Y X The resut [4, Coroary 1] aso shows X 4 1 W X 1+ X 4 for a X R This is inherited by W Young s and riedrichs inequaities yied λ f v L gv dx 1+ v L 1+ v L 4 1+ v W 1,4 for a v A, where L L 4 is a consequence of Höder s inequaity With ζx = λfx+gx +sup ξ>0 λξ ξ 3 we aso have DLx, ξ; = λξ fx gx ζx+ ξ 3 Hence the probem at hand fufis Assumptions 1 and By appying a Cauchy Schwarz inequaity to the definition of J, it foows that the continuous soution u of 13 and the discrete soutions u of 14 satisfy, for a v W 1,4, λ u u L Ju; u u Ju ; u u, Ju; v Ju ; v λ u u L v L This proves Assumptions 3 for λ > 0 or 4 for λ = 0, the atter up to condition 5, which is shown in the proof of [3, Lemma 91] for the probem at hand Together with the preceding resuts, this proves a the assumptions necessary in order to appy Theorem 44 Theorem Thereexistsatmostoneminimumu A W,p T 0 H 3/+ε with ε>0 Such a soution u satisfies u u H 1 = OH 3 A genera error estimate This section generaizes [3] from quasi-uniform trianguations to genera shape-reguar trianguations with proper weights Lemma 31 Let v W,t T 0 for some 1 <t< Then,w = v I v satisfies w L t + H w L t H v L t Copyright by SIAM Unauthorized reproduction of this artice is prohibited

7 STRONG CONVERGENCE IN CONVEX MINIMIZATION 4575 Proof The estimates in 1 with w = v ead to w t L t = w t L t T T v t L t Ht T v t L t, T T T T h t w t L t = T T w t L t T T T h t T v t L t T Ht v t L t Lemma 3 Let v H T 0 Then,w = v I v satisfies w H1+γ v L Proof The trace inequaity, 1, and h T h yied [ w ] L w T+ L + w T L h w L ω + h 1 w L ω h w + L ω h 1 h T w h v L ω L ω Reca that ω denotes the patch of, ie, the union set of the two trianges or tetrahedra having the common edge or face The definition of then gives w = ρ [ w ] L ρ h v L ω H1+γ v L Lemma 33 The discrete soutions are bounded in W 1,p, u W 1,p 1 Proof see [5, Lemma 41] With Assumption, the triange inequaity, and Lemmas 31 and 3, we have for w = u D u D,, Eu E u E u D, = W u D, x dx + Lx, u D, x dx + 1 u D, 1+ u D, p W 1,p + u D, q W 1,p + u D, 1+ u D p W 1,p + w p W 1,p + u D q W 1,p + w q W 1,p + w 1+H p + Hq + H1+γ 1 On the other hand, Assumption aso yieds Eu = W u x dx + u p W 1,p u q W 1,p 1 Lx, u x dx Hence there exists an -independent constant C>0 such that u p W 1,p u q W 1,p 1 C Since p>q, u W 1,p is bounded by the positive root of xp x q 1 C =0 Copyright by SIAM Unauthorized reproduction of this artice is prohibited

8 4576 WOLGANG BOIGER AND CARSTEN CARSTENSEN The boundedness of u W 1,p and riedrichs inequaity aso ead to the boundedness of u L p Remark 34 Simiar arguments can be used to prove the boundedness of any exact soution u, ie, sup u soves 11 u W 1,p < However, since we fix some soution u throughout the arguments, the anaysis focuses on the behavior of constants as The foowing resut is essentiay [3, Lemma 41] with the same outine of the proof but somehow different constants Lemma 35 Assumptions 1 and yied δ r L p c 1 δ : e dx, where p is the conjugate exponent of p, 1/p +1/p =1,andc 1 is given by c 1 =3 p s r p α p s p 1 + u p W 1,p + u p s p W 1,p, which is bounded with respect to Proof The proof foows that of [4, Theorem ] Define t := 1 + s/p Appying the tth root to shows δ r/t α 1/t 1 + u s + u s 1/t δ : e 1/t Integrating and appying Höder s inequaity with exponents t and t with 1/t+1/t = 1 resuts in δ r/t L r/t α1/t 1 + u s + u s 1/t δ : e 1/t dx α 1/t 1/t 1/t 1 + u s + u s t /t dx δ : e dx Here r/t > 1 is enforced by Assumption 1 With tp = st and L p 1 p t r L r/t a consequence of Höder s inequaity, this yieds s/p δ r L p r p t α 1 + u s + u s p/s dx δ : e dx Assumption 1 impies p>s The L p/s -norm on the right-hand side is estimated by appying another Höder s inequaity with exponents p/p s andp/s in R 3, ie, Therefore 1+ u s + u s = [1, 1, 1], [1, u s, u s ] 1 + u s + u s p s dx 3 p s s 3 p s p 1 + u p + u p s p + u p W 1,p + u p W 1,p Combining the ast inequaities yieds the caimed inequaity The constant c 1 is bounded due to Lemma 33 Copyright by SIAM Unauthorized reproduction of this artice is prohibited

9 STRONG CONVERGENCE IN CONVEX MINIMIZATION 4577 Lemma 36 see [3, Lemma 43] Let r > 0 be such that 1/r +1/r =1;then Assumptions 1 and yied with c = r c r 1 1 /r for every v V, δ : e v dx r 1 δ : e dx + c e v r W 1,p The preceding emma is proven in [3, Lemma 43] Lemma 37 see [3, Lemma 4] Assumption 3 or 4 impies for every v V, δ : e dx + 1 e + β δ : e v dx + M e L e v L + 1 e v The proof of the preceding emma is anaogous to the proof of [3, Lemma 4] Theorem 38 see [3, Theorem 1] With the conjugate exponent p of p, Assumptions 1 and together with Assumption 3 or 4 impy for every v V, 1 r 1 δ : e dx + c 1 1 δ r L p + e +β c e v r/r 1 W 1,p +M e L e v L + e v A terms on the eft-hand side are nonnegative The constants c 1,c > 0 as given in Lemmas 35 and 36 areboundedwithrespectto Proof This foows from the preceding emmas 4 Convergence resuts In the foowing we assume that Assumptions 1 and hod urther, we assume Assumption 3 or 4 to be true and that β and ζ are defined accordingy urthermore, we wi denote v = I e and w = e v Based on the two cases for β and ζ, our next step is proving the L -convergence of u Lemma 41 L -convergence Assumption 3 impies e L + e Hξ { for ξ := min 1+γ, Proof Theorem 38 and Young s inequaity ead to } r r 1 m e L + e c w r/r 1 W 1,p + m e L + M m w L + w We can appy the estimates provided by Lemmas 31 and 3 with v = e after subtracting m e L, which concudes the proof Lemma 4 L -convergence Assumption 4 impies e L + e Hξ for ξ := min {1+γ,} Proof As v = 0 on the boundary, a one-dimensiona riedrichs inequaity shows v L z v L, with z given in Assumption 4 This assumption Copyright by SIAM Unauthorized reproduction of this artice is prohibited

10 4578 WOLGANG BOIGER AND CARSTEN CARSTENSEN then yieds e L w L + v L w L + z v L w L + z e L + z w L w L + w L + δ : e We estimate the ast term with Theorem 38, which gives us e L w L + w L + w r/r 1 W 1,p + e L w L + w We absorb e L with the eft-hand side, which eaves another w L A remaining terms can be estimated with Lemmas 31 and 3 with v = e Observing r/r 1 proves the estimate on e L Using this and the above estimates in Theorem 38 again, together with Young s inequaity, we obtain e w r/r 1 W 1,p + e L + w L + w, which eads to the caimed estimate on e Coroary 43 uniqueness of the continuous soution The soution u given by 4 is unique Proof With Lemma 41 or 4, respectivey, the discrete soutions u L converge to the continuous soution u for H 0 This hods for a continuous soutions u, but the imit is unique The foowing main resut impies the H 1 error estimate u u H 1 = OH1/ for γ =1 Theorem 44 H 1 -convergence Either Assumption 3 or Assumption 4 impies { 1+γ e L Hξ/ for ξ =min,ζ 1+γ } Proof An integration by parts yieds e L = e : e dx T T T = T T T e νe ds = e νe ds e Δ e dx + [ e ] νe ds =: A + B + C T e Δe dx Estimate on A Since e = u D I u D on the boundary, we may appy 1 to each edge or face, which eads to e L h e L = h u D L Copyright by SIAM Unauthorized reproduction of this artice is prohibited

11 STRONG CONVERGENCE IN CONVEX MINIMIZATION 4579 Here u D denotes the n 1-dimensiona Hessian of u D on the face or edge We approximate the norma derivative e ν with the strong trace inequaity With T, this yieds e ν L e L e L T e L T + h 1 T u L T Summing up a and appying severa Cauchy inequaities proves A e L e ν L T T h u D L e 1/ L T e 1/ H u D L e L + H 3/ u D L e 1/ L u 1/ L T + h 1/ T L u 1/ L T Estimate on B Since u S 1 T, it hods that e = u Therefore B = e Δ e dx = e Δ u dx e L Estimate on C With ρ and a Cauchy inequaity, C = ρ 1/ [ e ] ν / e ds ρ 1/ [ e ] ν / L L e ρ [ e ] ν L e L = e e L The trace inequaity yieds e L max +max H 1 γ h e L ω + h 1 e L ω ρ 1 h ρ 1 h 1 e L ω e L ω e L + H 1 γ e L By combining the preceding estimates and absorbing e L and e 1/ L,we obtain e L H4 + H + e L + H1 γ e + H 1+γ e L e Copyright by SIAM Unauthorized reproduction of this artice is prohibited

12 4580 WOLGANG BOIGER AND CARSTEN CARSTENSEN Assumption 3 and Lemma 41 impy e L { ξ =min 4,, r r 1, 1+γ, Hξ/ r r 1 1+γ, with } r r 1 +1 γ Since 1+γ 1 γ and γ+1 r asweas r 1 = 1 r r 1 1+γ representation of ξ eads to the theorem Assumption 4 impies that the term which concudes the proof r r 1 +1+γ, this is repaced with, due to Lemma 4, REERENCES [1] S C Brenner and L Scott, The Mathematica Theory of inite Eement Methods, nded, Texts App Math 15, Springer, Berin, 00 [] C Carstensen and S Müer, Loca stress reguarity in scaar nonconvex variationa probems, SIAM J Math Ana, 34 00, pp [3] C Carstensen, P Pecháĉ, S Bartes, and A Proh, Convergence for stabiisation of degenerate convex minimsation probems, Interfaces ree Bound, 6 004, pp [4] C Carstensen and P Pecháĉ, Numerica soution of the scaar doube-we probem aowing microstructure, Math Comp, , pp [5] C Carstensen, Convergence of an adaptive EM for a cass of degenerate convex minimisation probems, IMA J Numer Ana, 8 008, pp [6] M Chipot, Eements of Noninear Anaysis, BirkhäuserAdv Texts 6, Birkhäuser, Base, 000 [7] P G Ciaret, The inite Eement Method for Eiptic Probems, SIAM, Phiadephia, 00 [8] B Dacorogna, Direct Methods in the Cacuus of Variations, nd ed, App Math Sci 78, Springer, Berin, 008 [9] S Müer, Variationa Modes for Microstructure and Phase Transisions, Cacuus of Variations and Geometric Evoution Probems, Lecture Notes in Math 1713, S Hidebrandt et a, eds, Springer, Berin, 1999, pp 85 10, ectures given at the nd session of the Centro Internazionae Matematico Estivo CIME, Cetraro, Itay, 1996 Copyright by SIAM Unauthorized reproduction of this artice is prohibited

13 CORRECTION TO ON THE STRONG CONVERGENCE O GRADIENTS IN STABILIZED DEGENERATE CONVEX MINIMIZATION PROBLEMS Because of a production error, section 4 in On the Strong Convergence of Gradients in Stabiized Degenerate Convex Minimization Probems by Wofgang Boiger and Carsten Carstensen is incorrect The correct version is as foows 4 Convergence resuts In the foowing we assume that Assumptions 1 and hod urther, we assume Assumption 3 or 4 to be true and that β and ζ are defined accordingy urthermore, we wi denote v = I e and w = e v Based on the two cases for β and ζ, our next step is proving the L -convergence of u Lemma 41 L -convergence Assumption 3 impies e L + e Hξ { for ξ := min 1+γ, Proof Theorem 38 and Young s inequaity ead to } r r 1 m e L + e c w r/r 1 W 1,p + m e L + M m w L + w We can appy the estimates provided by Lemmas 31 and 3 with v = e after subtracting m e L, which concudes the proof Lemma 4 L -convergence Assumption 4 impies e L + e Hξ for ξ := min {1+γ,} Proof As v = 0 on the boundary, a one-dimensiona riedrichs inequaity shows v L z v L, with z given in Assumption 4 This assumption then yieds e L w L + v L w L + z v L w L + z e L + z w L w L + w L + δ : e We estimate the ast term with Theorem 38, which gives us e L w L + w L + w r/r 1 W 1,p + e L w L + w We absorb e L with the eft-hand side, which eaves another w L A remaining terms can be estimated with Lemmas 31 and 3 with v = e Observing r/r 1 proves the estimate on e L Using this and the above estimates in Theorem 38 again, together with Young s inequaity, we obtain e w r/r 1 W 1,p + e L + w L + w, 1

14 ERRATUM which eads to the caimed estimate on e Coroary 43 uniqueness of the continuous soution The soution u given by 4 is unique Proof With Lemma 41 or 4, respectivey, the discrete soutions u L converge to the continuous soution u for H 0 This hods for a continuous soutions u, but the imit is unique The foowing main resut impies the H 1 error estimate u u H1 = OH1/ for γ =1 Theorem 44 H 1 -convergence With ξ from Lemma 41 respectivey, Lemma 4, Assumption 3 or Assumption 4 impies { ξ e L Hη for η =min 4, ξ 1+γ } 4 Proof An integration by parts yieds e L = e : e dx T T T = T T T e νe ds = e νe ds e Δ e dx + [ e ] νe ds =: A + B + C T e Δe dx Estimate of A Since e = u D I u D on the boundary, we may appy 1 to each edge or face, which eads to e L h e L = h u D L Here u D denotes the n 1-dimensiona Hessian of u D on the face or edge We approximate the norma derivative e ν with the strong trace inequaity With T, this yieds With T, this yieds e ν L e L e L T u L T + h 1 T e L T The sum over a and severa Cauchy inequaities prove A e L e ν L T T H 3/ + H h u D L e 1/ L T u 1/ + L T h 1/ T e 1/ L T u D L e L u D L e 1/ L u 1/ L

15 ERRATUM 3 Estimate of B Since u S 1 T, it hods e = u Therefore B = e Δ e dx = e Δ u dx e L Estimate of C With ρ and a Cauchy inequaity, C = ρ 1/ [ e ] ν / e ds ρ 1/ [ e ] ν / L L e ρ [ e ] ν L e L = e e L The trace inequaity yieds e L max +max H 1 γ h e L ω + h 1 e L ω ρ 1 h ρ 1 h 1 e L ω e L ω e L + H 1 γ e L By combining the preceding estimates and absorbing e L and e 1/ L,we obtain e L H4 + H + e L + H1 γ e 1+γ + H e L e Lemma 41 or 4 with a different ξ impies e L Hη { 3 η =min, 4 3, ξ 4, ξ + 1 γ, ξ 1+γ } 4 with Since ξ 4 1+γ theorem and 1+γ 4 < 1 γ this representation of η eads to the