The inuene of quadrature formulas in 2Dand 3D mortar element methodsYvon Maday1, Franesa Rapetti1, and Barbara I. Wohlmuth21 Laboratoire d'Analyse Num�erique, Paris 6 University, Bô�te ourrier 187,75252 Paris edex 05, Frane2 Mathematishes Institut A, Universitat Stuttgart, Pfa�enwaldring 57,70 569 Stuttgart, GermanyAbstrat. The paper is onerned with the mortar �nite element disretizationof salar ellipti equations in three dimensions. The attention is foused on theinuene of quadrature formulas on the disretization error. We show numeriallythat the optimality of the method is preserved if suitable quadrature formulas areused.1 IntrodutionThe mortar element method, �rstly proposed in [3℄, is a non-onforming non-overlapping domain deomposition method. The oupling of di�erent phys-ial models, disretization shemes or non-mathing triangulations at theinterfaes an be eÆiently realized in terms of mortar element methods. Topreserve the global optimality of the loally adapted disretizations, the in-terfaes between the di�erent regions have to be handled appropriately. Dueto its high exibility, this approah has been analysed and implemented inmany situations.The main feature of the mortar element methods is to replae the exatontinuity ondition at the skeleton of the deomposition with a weak one.An important aspet for the implementation is the realization of the weakoupling aross the interfaes. It an be written in terms of a Lagrange mul-tiplier and the jump of the traes. The assoiated integrals involve disretefuntions de�ned on di�erent non-mathing grids and, as a onsequene, theomputation goes through the intersetion of the supports. In two dimen-sions, for �nite element disretizations, the supports an be easily intersetedat the ommon interfaes due to the fat that the latter are one dimensionalurves. In three dimensions, the intersetion of the supports (union of trian-gles or quadrilaterals) beomes a hard task. The use of quadrature formulasto evaluate these integrals inreases onsiderably the eÆieny of the imple-mentation. However, using a standard Galerkin approximation and replaingthe exat integration by a quadrature formula based only on the slave ormaster side does not yield optimal results (see [5℄ for numerial evidenes intwo dimensions). The best approximation error requires a quadrature formulabased on the slave side and the onsisteny error one on the master side. As a

2 Yvon Maday, Franesa Rapetti, Barbara I. Wohlmuthonsequene, we are led to work with di�erent test and trial spaes, resultingin a Petrov-Galerkin approah. We fous our attention on the inuene ofquadrature formulas on the disretization error. Numerial results are pre-sented for a salar ellipti equation in two and three dimensions. Beware ofthe diÆulty of the proof in three dimensions, we show only numerially thatthe optimality of the method is preserved if suitable quadrature formulas areused. We are aware that few examples are not suÆient to onlude aboutthis subjet but they give a �rst insight on it. Asymptotially, the error inthe energy norm is O(h) and O(h2) for the L2-norm.2 Problem formulation and notationThe key example for the de�nition and the analysis of the mortar elementmethod is the following seond order ellipti boundary value problem� div (a grad u) = f in ;u = 0 on �D ;� u� n = 0 on �N ; (1)where a is a suÆiently smooth matrix that is uniformly positive de�nite inthe bounded open set � IR3, �D is a subset of the boundary � of withmeas(�D) > 0 and �N = � n �D.Let us introdue the funtional spaeH10;D() = fu 2 H1() juj�D = 0 g,that is the losure in H1() of all C1-funtions vanishing on �D . Then,problem (1) admits the following variational formulation:Given f 2 L2() ; �nd u 2 H10;D() suh thatR a grad u � grad v d = R f v d 8 v 2 H10;D(): (2)Problem (2) admits a unique solution, due to the Lax-Milgram theorem andthe Poinar�e lemma.Suppose now that is deomposed into non-overlapping subdomains k,k = 1; : : : ;K, suh that = [Kk=1k ; (k \` = ;k \` = � k` 8 k 6= ` : (3)We introdue the skeleton S of the deompositionS = [k;`�k` : (4)�k`; �N and �D will be assumed to be the union of polygonal subsets ofthe boundaries of the subdomains k: often, suh a deomposition is alled

The inuene of quadrature formulas in mortar element methods 3geometrially onforming. To de�ne orretly the trae spae, we assoiatewith eah interfae �k` the non-mortar (slave) side whih is, by onvention,k whereas ` is the mortar (master) side. Let B denote a bounded opensubset of IR3. If 0 is a smooth subset of �B, the spae H1=200 (0) onsists ofthose elements v 2 H1=2(0) whose trivial extension ~v of v by zero to all �Bbelongs to H1=2(�B) [6℄, i.e.H1=200 (�k`) = fv 2 H1=2(�k`) j ~v 2 H1=2(�k) g ;jjvjjH1=200 (�k`) = jj~vjj1=2;�k ; (5)where the norm of ~v is evaluated on the side �k. We �nally remind thatH1=200 (�k`) is a proper and ontinuously embedded subspae of H1=2(�k`) [6℄.In the following, (H1=200 (�k`))0 denotes the dual spae of H1=200 (�k`). We nowreturn to the haraterization of H10;D(). To this purpose, we introdue thespae X� = fv 2 L2() j vjk 2 H1(k) ; k = 1; : : : ;K ; vj�D = 0 g ; (6)endowed with the broken normjjvjj� = KXk=1 jjvjj21;k!1=2 ; (7)and a proper subspaeX00 = fv 2 X� j [v℄j�k` 2 H1=200 (�k`) ; 8�k` � S g (8)where [v℄j�k` is the restrition of the jump (vjk � vj`), for v 2 X�, to anyinterfae �k` � S. Then, the following spaeV = fv 2 X00 j < �; [v℄ >0;�k`= 0 ; 8� 2 (H1=200 (�k`))0 ; �k` � S gis equal to H10;D(). Problem (2) an be written into its equivalent domaindeomposition formulation:Given f 2 L2() ; �nd u 2 V suh that 8 v 2 VPKk=1 Rk a grad uk � grad vk d =PKk=1 Rk f vk d : (9)As a onsequene, problem (9) admits a unique solution.3 Problem disretizationIn eah subdomain k, we hoose a family of onforming triangulations(Tk;h)h, independently of the ones de�ned in the neighboring subdomains,

4 Yvon Maday, Franesa Rapetti, Barbara I. Wohlmuthi.e. the nodes in Tk;h that belong to �k` do not need to math the nodesof T`;h, k 6= `. We onsider triangulations omposed of tetrahedra t and wede�ne the loal disrete spaes of pieewise linear �nite elements on Tk;h suhas Xk;h = fvk;h 2 H1(k) j vk;hjt 2 IP1(t) ; 8 t 2 Tk;h ; vk;hj�D\�k = 0 g :We set Xh = �Kk=1Xk;h = fvh 2 L2() j vk;h = vhjk 2 Xk;hg :A partiular attention has to be addressed to the trae of elements of Xh onthe interfae �k` between adjaent subdomains k and `. We denote asTk`;h = fw 2 L2(�k`) j 9 vk;h 2 Xk;h suh that vk;hj�kl = w gthe spae of all ontinuous pieewise linear funtions on �k` on the partitionindued by the triangulation Tk;h of k. Note that Tk`;h 6= T`k;h sine thetriangulations Tk;h and T`;h do not math at the interfae �k`. In the mortarsetting, the value of eah disrete funtion on the mortar side will de�neweakly the values of the disrete funtion on the non-mortar side. We assoiatea loal spae Mk`;h to eah non-mortar side �k` of the skeleton with thefollowing features (that haraterizes the mortar element methods amongother hybrid formulations):(i) Mk`;h � Tk`;h ;(ii) dim Mk`;h = dim (Tk`;h \H10 (�k`)) ;(iii) Mk`;h ontains onstants on �k` : (10)We introdue the spae Mh of Lagrange multipliers on the skeleton S asMh = ��k`�SMk`;h : (11)The hoie of the Lagrange multiplier spae Mh is of key importane for theoptimality of the method. If we setuh = (uk;h)k ; mh = (�k`)�k`�Sa(uh; vh) =PKk=1 Rk a grad uk;h � grad vk;h db(vh;mh) =P�k`�S R�k` (vk;h � v`;h)�k` d�the disrete problem readsGiven f 2 L2() ; �nd uh 2 Vh suh that 8 vh 2 Vha(uh; vh) =PKk=1 Rk f vk;h d (12)

The inuene of quadrature formulas in mortar element methods 5where Vh is the onstrained spaeVh = fvh 2 Xh j b(vh;mh) = 0 ; 8mh 2Mh g :Note that in this way the Lagrange multipliers are not unknowns of theproblem sine we work with test and trial funtions that already satisfy theweak oupling ondition ontained in Vh. The presene of this weak ouplingondition prevents Vh from being a subspae of V , i.e. we are using a non-onforming method to approximate the solution of problem (9). The seondStrang lemma allows to derive the following error bound for suh an approx-imationjju� uhjj1;� � ( infvh2Vh jju� vhjj1;� + supwh2Vh P�k`�S R�k` a �u�n [wk;h℄ d�jjwhjj1;� ) : (13)In the right-hand side of (13), the �rst term represents the best approximationerror and the seond is the onsisteny error.In the following setions we address two diÆulties: the �rst is the on-strution of a basis for the Lagrange multiplier spae Mh de�ned in (11) andthe seond is the one enountered to satisfy the oupling ondition ontainedin Vh, i.e. 8�k` � S and 8�k` 2Mk`;h,Z�k` (vk;h � v`;h)�k` d� = 0 ; vk;h 2 Xk;h ; v`;h 2 X`;h: (14)3.1 A basis for Mk`;hWe now detail the andidate proposed in [1℄, [2℄, [3℄ for Mk`;h satisfying therequirements (i); (ii); (iii) listed in the previous setion. To this purpose, we�x some notations. Let � = �k` and Tk;� is the triangulation in triangles tthat is indued on � by the partitioning Tk;h of the non-mortar side k. LetV� ; V� and V�� denote the set of all nodes of Tk;� , the nodes inside � andthose on the boundary of � , respetively. Note that V� = V� [ V�� . The�nite element basis funtions will be denoted by 'a; a 2 V� . We de�ne~X�k;h = span f'a j a 2 V� g�a = supp ('a) = [ft 2 Tk;� j a vertex of t gNa = fb 2 V� j b 2 �a g ; N = [a2V�� Nawhere Na is the set of the internal neighboring nodes of a and N is the setof those internal nodes whih have a neighbor on the boundary of � . Now,we set Mk`;h = span f'̂a j a 2 V� gwhere the basis funtions '̂a are de�ned as follows:'̂a = ('a ; a 2 V� n N ;'a +Pb2V��\�a b;a 'b +Pd2V�� d;a 'd ; a 2 N :

6 Yvon Maday, Franesa Rapetti, Barbara I. WohlmuthTo be more preise, we set b;a = 1=nb where nb is the number of nodesin Nb and the oeÆient d;a = 1 if d is the node opposite to a and zerootherwise. We all d 2 V�� opposite node to a 2 V� if ��a \ ��d is one edgeand Nd is empty. For more sophistiated hoies of b;a and d;a, we refer to[4℄. Note that '̂a(b) = Æa;b, for all a; b 2 V� , i.e. the funtions '̂a, a 2 V� , arelinearly independent. Requirements (i) and (ii) are satis�ed by onstrution.Observing that, for all b 2 V�� having no opposite node,Xa2Nb b;a = 1holds and that Xd2V�� ;d opp. to a2V� d;a = 1 ;we �nd that (iii) is satis�ed. Moreover, we obtain that Pa2V� '̂a = 1 :As an example, we present a basis for the Lagrange multiplier spaeMk`;hwhere Tk;� is given in Figure 1. In this ase, node 1 is opposite to node 12so 1;12 = 1 whereas 1;i = 0 for all other nodes i 6= 12. Note that all othernodes on the boundary have no opposite node.1 2 3

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Fig. 1. Example of triangulation on a non-mortar side.We have V�� = f1 ; : : : ; 11 g ; V� = f12 ; : : : ; 21 g ;N1 = ; ; N2 = f12; 13g ; N13 = f12; 15; 19g ; N = V� n f15; 21g ;and the basis funtions for Mk`;h are'̂12 = '12 + '1 + 12'2 + 12'11 ; '̂13 = '13 + 12'2 + '3 + 13'4 ;'̂14 = '14 + 13'10 + 12'11 ; '̂15 = '15 ;'̂16 = '16 + 13'8 + 13'10 ; '̂17 = '17 + 13'4 + '5 + '6 + 12'7 ;'̂18 = '18 + 12'7 + 13'8 ; '̂19 = '19 + 13'4 ;'̂20 = '20 + 13'8 + '9 + 13'10 ; '̂21 = '21 :

The inuene of quadrature formulas in mortar element methods 7In the following, we denoteVS = [�k`�S V�k` ; V�S = [�k`�S V��k` ; VS = [�k`�S V�k` ;where the unions are intended without repetitions and VS = VS [ V�S .3.2 Evaluation of the integrals on the interfaes �k` � SCondition (14) is involved in the de�nition of a basis for the disrete on-strained spae Vh. The omputation of the quantity R�k` vk;h �k` d� raises nodiÆulty sine the two disrete funtions vk;h and �k` live on the same non-mortar mesh inherited from k on �k`. On the ontrary, the omputation ofR�k` v`;h �k` d� involves disrete funtions that live on di�erent meshes sinev`;h is de�ned on the master mesh inherited from ` on �k` that may notmath with the one where �k` is de�ned, as it is shown in Figure 2. To avoidinterseting the supports, a task that an be rather diÆult and expensivefor interfaes of ompliated shape and �nite urvature, we resort to a moreeÆient tehnique involving lassial numerial quadratures.������������������������������������

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quadrature pointsmaster pointsFig. 2. Example of domain deomposition into two subdomains and non-mathinggrids on the skeleton of the deomposition.In order to simplify the presentation, we fous our attention again on oneinterfae �k` � S. We denote by �+ (resp. v+) the mortar (master) sideof �k` (resp. the value of v on the mortar side) and by �� (resp. v�) thenon-mortar (slave) side of �k` (resp. the value of v on the non-mortar side).Numerial quadratures lead to approximate the integrals on the mortar (resp.on the non-mortar) side as followsR�+ w+d� �P+ w+ � resp. R�� w�d� �P� w��where P+ (resp. P�) represents the numerial integration on the mortar(resp. non-mortar) side. Condition (14) an be then rewritten as followsZ�� (v� � v+)�� d� = 0 ; 8�� 2 S ; 8�� 2M�;h :

8 Yvon Maday, Franesa Rapetti, Barbara I. WohlmuthSine v� and �� are de�ned on the slave side, it is natural to apply a quadra-ture formula on this side to ompute R�� v� �� d� . We use then a quadratureformula whih is exat, i.e. R�� v� �� d� =P� v� �� :To evaluate R�� v+ �� d� , there are two natural possibilities. We de�netwo disrete spaes V �+h and V ��h byV �+h = fvh 2 Xh j P� v�h �m�h =P+ v+h �m�h ; 8mh 2Mh g ;V ��h = fvh 2 Xh j P� v�h �m�h =P� v�h �m�h ; 8mh 2Mh g :With these two hoies, problem (12) reads respetivelygiven f 2 L2() ; �nd uh 2 V �+h suh that 8 vh 2 V �+ha(uh; vh) =PKk=1 Rk f vk;h d ; (15)given f 2 L2() ; �nd uh 2 V ��h suh that 8 vh 2 V ��ha(uh; vh) =PKk=1 Rk f vk;h d : (16)In [5℄, the authors have shown that, in two dimensions, the Galerkin approahto solve (12) in V �+h or in V ��h leads to a loss of a

uray. In partiular,V �+h is good for the onsisteny error and not for the best approximationone, whereas V ��h does exatly the opposite. To maintain the optimality ofthe non-onforming approximation, we adopt a Petrov-Galerkin approah, byhoosing a test spae whih is di�erent from the trial spae. Then problem(12) beomesgiven f 2 L2() ; �nd uh 2 V ��h suh that 8 vh 2 V �+ha(uh; vh) =PKk=1 Rk f vk;h d : (17)Unfortunately, we annot prove yet the well-posedness of problem (17). How-ever, our numerial results show it is the ase and they yield an optimalsheme.Remark 1. The hoie of the test and trial spaes an be motivated by thefollowing interpretation. In the mortar approah, we have to solve a oupledDirihlet-Neumann problem. On the slave side, a Dirihlet problem has tobe onsidered where the boundary ondition is obtained from the trae onthe master side. As it an be done in the ontext of inhomogeneous Dirihletboundary onditions, we realize the boundary ondition in a weak integralform based on quadrature formulas on the mesh whih is the mesh on the slaveside. Thus the natural hoie for the trial spae is V ��h . On the other hand,on the master side, we solve a Neumann problem where the inhomogeneousboundary onditions are obtained from the residual on the slave side. Inthis ase, a quadrature formula on the master side is the natural hoie.Neumann boundary onditions enter in a weak form on the right-hand side.More preisely, they are seen by the test spae. Thus V �+h is the naturalhoie for the test spae.

The inuene of quadrature formulas in mortar element methods 94 Matrix form of the disrete problemTo underline the di�erene with a Galerkin (symmetri) approah, we writethe three disrete problems (15), (16) and (17) in matrix form. We start byonsidering on eah subdomain the unonstrained sti�ness matrix assoiatedwith homogeneous Neumann type onditions on any interfae � � S (thesystem is the same in the three ases). In the next step, the loal systems areoupled by means of the mortar ondition and this will make the di�erenebetween the Galerkin and the Petrov-Galerkin approahes.Let uh be the solution of the disrete problem: we expand eah of itsomponents uk;h in terms of the basis funtions of the �nite dimensionalspae Xk;h de�ned in the subdomains. We then have mk oeÆients uks , s =1; : : : ;mk, where mk is the dimension of Xk;h. Note that, due to the ouplingondition, only mk�nk` oeÆients are real unknowns of the problem in k,where nk` is the number of nodes on the non-mortar interfaes �k` of �k.6

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4 3Fig. 3. Let us onsider a subdomain k and let us suppose that � 1456 = �D\�k,for example. We assume that �1276 and �5678 are mortar interfaes whereas �1234,�2387 and �4385 are non-mortar interfaes. The real unknowns in k are the valuesassoiated to nodes in k n (�1234 [ �2387 [ �4385 [ � 1456).For eah interfae �k` � S \ �k, we haveC1 uk;�V�k\�k` = D1 u`;+V�`\�k` +D2 u`;+V�`\��k` � C2 uk;�V�k\��k`where C1(i; j) = R�k` '�j '̂�i d� ; 8 i 2 V�k\�k` ; 8 j 2 V�k\�k` ;C2(i; j) = R�k` '�j '̂�i d� ; 8 i 2 V�k\�k` ; 8 j 2 V�k\��k` ;D1(i; j) = R�k` '+j '̂�i d� ; 8 i 2 V�k\�k` ; 8 j 2 V�`\�k` ;D2(i; j) = R�k` '+j '̂�i d� ; 8 i 2 V�k\�k` ; 8 j 2 V�`\��k` :Now, we denoteC�1 (i; j) =P� '�j '̂�i ; 8 i 2 V�k\�k` ; 8 j 2 V�k\�k` ;C�2 (i; j) =P� '�j '̂�i ; 8 i 2 V�k\�k` ; 8 j 2 V�k\��k` :

10 Yvon Maday, Franesa Rapetti, Barbara I. Wohlmuthand D�1 (i; j) =P� '+j '̂�i ; 8 i 2 V�k\�k` ; 8 j 2 V�`\�k` ;D�2 (i; j) =P� '+j '̂�i ; 8 i 2 V�k\�k` ; 8 j 2 V�`\��k` ;D+1 (i; j) =P+ '+j '̂�i ; 8 i 2 V�k\�k` ; 8 j 2 V�`\�k` ;D+2 (i; j) =P+ '+j '̂�i ; 8 i 2 V�k\�k` ; 8 j 2 V�`\��k` :Note that when the value of a funtion v+ is needed in a point x� of a grid T �whih is di�erent from the grid T + where the funtion v+ lives, we projetx� on T + and we ompute v+ in the projetion point. We denote, in a bloklayout, Q+k` = [(C�1 )�1D+1 ; (C�1 )�1D+2 ; �(C�1 )�1 C�2 ℄ ;Q�k` = [(C�1 )�1D�1 ; (C�1 )�1D�2 ; �(C�1 )�1 C�2 ℄ :Then the interfae ondition for vh 2 V �+h on the interfae �k` readsvk;�V�k\�k` = Q+k` [v`;+V�`\�k` ;v`;+V�`\��k` ;vk;�V��k`\��k` ℄ ; (18)and for vh 2 V ��h on the same interfae �k` readsvk;�V�k\�k` = Q�k` [v`;+V�`\�k` ;v`;+V�`\��k` ;vk;�V��k`\��k` ℄ : (19)In the non-symmetri approah we have the relation (19) for the trial fun-tions and the relation (18) for the test funtions.Remark 2. We note that in the implementation, the matrix (C�1 )�1 will benever assembled. To ompute the ation of (C�1 )�1 on a vetor, one has tosolve a mass matrix system per interfae. If dual Lagrange multipliers [7℄ areused, the algorithm is even more eÆient sine the matrix C�1 is diagonaland its inversion is for free.If A denote the blok sti�ness matrix, the matrix form is(Q+)tAQ+U = (Q+)t F ; for problem (15) ;(Q�)tAQ�U = (Q�)t F ; for problem (16) ;(Q+)tAQ�U = (Q+)t F ; for problem (17) :Note that Q+ and Q� are blok matries built from Q+k` and Q�k`. Withinthe Galerkin approahes (15) and (16), the �nal system has a symmetri andpositive de�nite matrix. The system an be easily solved by a preonditionedConjugate Gradient method. We note that there is a unique oupling matrixto ompute but the results with these approahes are not optimal. Withinthe Petrov-Galerkin approah, the �nal system has a non-symmetri matrix.In this ase, we use a Bi-Conjugate Gradient method. We note that there are

The inuene of quadrature formulas in mortar element methods 11two oupling matries to ompute but the results within this approah areoptimal.Thanks to the mortar element method philosophy, in the adopted itera-tive proedure, residuals an be omputed in parallel as observed in [1℄. Inpartiular, sine we do not want to deal with the assembled matrix QtAQ,we have to work with the \objets" Qk`, Ak (k; ` = 1; : : : ;K) separately. Fora vetor v 2 IRn with n =PKk=1 mk �P�k`�S nk`, where nk` is the numberof nodes in the interior of the non-mortar side �k`, the matrix-vetor produtQtAQv, where Q is Q+ or Q� a

ording to the system we solve, an be donein three steps:(step1) we ompute q = Qv with q 2 IRs with s =PKk=1 mk, separately oneah non-mortar interfae �k`;(step2) we ompute p = Aq separately in eah subdomain k;(step3) we ompute w = Qtp with w 2 IRn, separately on eah non-mortarinterfae �k`.In this three-step produt, the most expensive step is the seond one dueto the size of the subdomain matries; the produts involving the ouplingmatrix are not expensive at all (the number of nodes, and onsequently of theunknowns, on S is inferior to that in the adjaent volumes and the Lagrangemultiplier supports on S are rather small). As a onsequene, the iterativesolution of the system in the three ases is a

omplished with a sensible gainin time and without involving additional memory spae.5 Numerial results in 3DWe onsider the on�guration presented in Figure 4 (left), where = 1 [2 = (0; 2)� (0; 1)� (0; 1) ; � = f(1; y; z) j y; z 2 (0; 1) g ;�D = ABCD [ EFGH ; �N = � n �D :Sine �� \ ��N 6= ;, we have a deomposition with \ross-points", i.e. theunknowns that are assoiated to all the points on �� \ ��N , even if theybelong to the slave side, they are real degrees of freedom.The right hand side f and the boundary onditions for ��u = f arehosen suh that the exat solution is given byu(x; y; z) = os(� y) os(� z) [2x� x2 + sin(� x) ℄and in Figure 4 (right) we present its behavior on the interfae � . The normalderivative on the interfae of the analytial solution is given by�nu(y; z) = �� os(� y) os(� z)where n = (1; 0; 0)t and its value will be used later to evaluate the onsis-teny error on the slave part. We note that the disrete Lagrange multiplier

12 Yvon Maday, Franesa Rapetti, Barbara I. WohlmuthA

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Fig. 4. Computational on�guration and analytial solution on the interfae.approximates the ux of the solution aross the interfae � . We use low orderquadrature formulas. On the referene triangle of verties (0; 0); (1; 0); (0; 1),we take the quadrature points ( 23 ; 16 ); ( 16 ; 16 ); ( 16 ; 23 ) with weights equal to 13 .Table 1. Error in L2 norm with two ubes. A Petrov-Galerkin approah is adopted.Nodes Error in 1 Error in 2 Global error125/64 0.10649 0.04576 0.11591343/125 0.06009 0.03183 0.068011000/512 0.02954 0.01418 0.032762197/1331 0.01750 0.08101 0.01928The global error in the L2 norm is presented in Figure 5 together withthe asymptoti order O(h2) and the detail is given in Table 1. As it an beremarked, the omputed error is in good agreement with the theoretial one(the two slopes are parallel).Table 2. Error inH1 norm with two ubes. A Petrov-Galerkin approah is adopted.Nodes Error in 1 Error in 2 Global error125/64 1.10769 0.61424 1.26660343/125 0.81070 0.50506 0.955161000/512 0.57225 0.32739 0.659282197/1331 0.44235 0.24082 0.50366The global error in the H1 norm is presented in Figure 6 together withthe asymptoti order O(h) and the detail is given in Table 2. Again, theomputed error is in good agreement with the theoretial one.

The inuene of quadrature formulas in mortar element methods 130.007

0.015

0.03

0.07

0.15

189 468 1512 3528

Err

or in

the

L2 n

orm

Number of Nodes

non-symmO(h^2)

Fig. 5. Global error in L2 norm with two ubes adopting a Petrov-Galerkin ap-proah.0.3

0.5

0.7

1.0

1.5

189 468 1512 3528

Err

or in

the

H1

norm

Number of Nodes

non-symmO(h)

Fig. 6. Global error in H1 norm with two ubes adopting a Petrov-Galerkin ap-proah.In Figures 7 and 8, we present the numerial solution on the interfae �omputed on the master and slave sides disretized with non-mathing gridswhen a Petrov-Galerkin approah is onsidered. Note that already with fewmesh nodes per subdomain, the omputed solution shown in Figure 7 is loseto the analytial one presented in Figure 4 (right); as long as we inrease thenumber of mesh nodes, the two omputed funtions given in Figure 8 beomeloser to the analytial one.Now we hek, on the interfae � , that the Galerkin approah for problem(12) gives a bad approximation error when Vh = V �+h and a bad onsistenyerror when Vh = V ��h . For the best approximation error on the interfae,of asymptoti order O(h3=2), we ompare the Petrov-Galerkin approah andthe Galerkin one with Vh = V ��h in Figure 9 and Vh = V �+h in Figure10, respetively. As predited, the Galerkin approah with Vh = V �+h is notoptimal for what onerns this error.

14 Yvon Maday, Franesa Rapetti, Barbara I. Wohlmuth−1

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Fig. 7. Slave and master solutions with resp. 49 and 25 mesh points on � .−1

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Fig. 8. Slave and master solutions with resp. 149 and 121 mesh points on � .0.007

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216 1050 2649 5346 9438 18816

Err

or in

the

L2 n

orm

Number of Elements

non-symmsymm quad -

O(h^(3/2))

Fig. 9. Best approximation error on � for the Galerkin approah with Vh = V ��hand the Petrov-Galerkin one.For the onsisteny error on the interfae, of asymptoti order O(h1=2), weompare the Petrov-Galerkin approah and the Galerkin one with Vh = V ��hin Figure 11 and Vh = V �+h in Figure 12, respetively. As predited, theGalerkin approah with Vh = V ��h is not optimal for this error.

The inuene of quadrature formulas in mortar element methods 150.007

0.015

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0.07

0.15

216 1050 2649 5346 9438 18816

Err

or in

the

L2 n

orm

Number of Elements

non-symmsymm quad +

O(h^(3/2))

Fig. 10. Best approximation error on � for the Galerkin approah with Vh = V �+hand the Petrov-Galerkin one.0.3

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1.11.31.5

216 1050 2649 5346 9438 18816

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sist

ency

err

or in

the

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orm

Number of Elements

non-symmsymm quad -

O(h^(1/2))

Fig. 11. Consisteny error on � for the Galerkin approah with Vh = V ��h and thePetrov-Galerkin one.0.3

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216 1050 2649 5346 9438 18816

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err

or in

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orm

Number of Elements

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Fig. 12. Consisteny error on � for the Galerkin approah with Vh = V �+h and thePetrov-Galerkin one.

16 Yvon Maday, Franesa Rapetti, Barbara I. Wohlmuth6 Numerial results in 2DIn this setion, we present some numerial results in two dimensions obtainedby using the mortar element method with Lagrange multiplier spaes basedon a dual basis [7℄. In suh a ase, a biorthogonality relation between the nodalbasis funtions of these spaes and the �nite element trae spaes holds. Amain advantage of these Lagrange multiplier spaes is that the loality ofthe support of the nodal basis funtions of the onstrained spae an bepreserved.Analytical solution

u(x,y)

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x

Normal derivative on the interface

dn_u(x)

Fig. 13. Analytial solution and its normal derivative on the interfae.The unit square is deomposed into two retangles 1 := (0; 1)� (0; 0:5)and 2 := (0; 1)�(0:5; 1). The right hand side f and the boundary onditionsfor ��u = f are hosen suh that the exat solution, presented in Figure 13(left), is given by u(x; y) = (1� x) os(50(x� 12)y) :Then, the disrete Lagrange multiplier is an approximation of the ux arossthe ommon interfae � = f(x; 0:5) jx 2 (0; 1) g that is�nu(x) = �50(1� x)(x � 12) sin(25(x� 12))whih is given as a funtion of x in Figure 13 (right). Here, we have �xed thenormal diretion n = (0; 1)t. In Figure 14 we present the adopted deompo-sition together with the oarse mesh and the numerial solution omputedon the �nest mesh with our Petrov-Galerkin approah. At eah level, thequadrilaterals of the urrent mesh are divided by 4 up to a level where theglobal number of elements is 688128.As in the 3D ase, we use low order quadrature formulas. On the refereneinterval [0; 1℄, we take the three Gaussian quadrature points 0:11270167, 0:5,0:88729833 and the orresponding weights 5=18, 4=9, 5=18. In the following

The inuene of quadrature formulas in mortar element methods 17Fig. 14. Coarse mesh of quadrilaterals in the two-domain on�guration and nu-merial solution omputed on the �nest grid omposed of 688128 elements.�gures, the dotted line indiates the asymptotial order of the mortar elementmethod. The solid line is obtained in the non-symmetri ase and the dottedline with stars is obtained in the symmetri ase if the quadrature formula isapplied only on the master side. The three �gures are onneted with the errorin the energy norm, L2 norm and the mesh dependent Lagrange multipliernorm, the latter given by (Pe2� he jj�� � � u� n jj20;e)1=2, where e denotes anedge on the slave side of � . The non-symmetri Petrov-Galerkin approahyields in the energy and L2 norm asymptotially the same order as preditedby the theory for exat integration. This is not the ase for the symmetriapproah with Vh = V �+h .

0.001

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O(h)

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Err

or in

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orm

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O(h^2)

Fig. 15. Relative error in energy (left) and L2 (right) norm for the Galerkin ap-proah with Vh = V �+h and the Petrov-Galerkin one.In Figure 16, we show the omputed errors in the mesh dependent La-grange multiplier norm. From the point of theory, we expet to �nd an O(h)behavior. However, we observe asymptotially O(h3=2). This orresponds tothe best approximation error of the Lagrange multiplier spae and an beshown under the assumption that the error in the energy norm is asymptot-ially equally distributed.

18 Yvon Maday, Franesa Rapetti, Barbara I. Wohlmuth0.0001

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100 1000 10000 100000 1e+06E

rror

in th

e w

eigh

ted

LM n

orm

Number of Elements

non-symm.symm. Quad

O(h^(3/2))O(h)

Fig. 16. Error in the mesh dependent Lagrange multiplier norm for the Galerkinapproah with Vh = V �+h and the Petrov-Galerkin one.In Table 3, we report the error in the energy and L2 norm within the exatintegration and Petrov-Galerkin approahes. We remark that the values inolumns 2 and 4 are very lose to the one in olumns 1 and 3, stating oneagain the optimality of the non-symmetri approah. There is no signi�antdi�erene in the obtained a

uray. The Petrov-Galerkin approah whih isonsiderably heaper gives the same qualitative and quantitative results asthe symmetri mortar based on exat integration.Table 3. Relative error for exat integration and Petrov-Galerkin approahex. int. L2 norm P.-G. L2 norm ex. int. energy norm P.-G. energy norm1.564561e+00 1.590183e+00 1.021536e+00 1.022896e+003.867007e-01 3.891088e-01 6.427550e-01 6.430641e-011.087453e-01 1.087666e-01 3.283115e-01 3.283159e-012.813904e-02 2.813497e-02 1.667178e-01 1.667166e-017.097746e-03 7.133550e-03 8.364393e-02 8.364368e-021.778188e-03 1.788415e-03 4.185135e-02 4.185134e-024.447664e-04 4.485467e-04 2.092929e-02 2.092929e-021.112059e-04 1.114683e-04 1.046497e-02 1.046496e-02Finally, Table 4, we ompare the error in the weighted Lagrange mul-tiplier norm for the four di�erent approahes. Note that the one based onexat integration is the most expensive one. The results for the symmetriapproah with Vh = V �+h are not good and this is on�rmed by the valuesdisplayed in olumn 2 of Table 4. On the other hand, the values in olumn3 of Table 4 for the symmetri approah with Vh = V ��h are rather goodbut less than the ones in olumn 4 of the same table, that are related to

The inuene of quadrature formulas in mortar element methods 19the Petrov-Galerkin approah. As a onsequene, it an be remarked thatapplying the quadrature rule only on the slave side gives surprisingly goodresults for this example. More numerial tests have to be arried out to il-lustrate the di�erene between the symmetri approah with Vh = V ��h andthe Petrov-Galerkin approah. It seems that the Lagrange multiplier norm isthe most sensitive norm to measure the inuene of the onsisteny error.Table 4. Error in the weighted Lagrange multiplier norm.Exat integration Galerkin (V �+h ) Galerkin (V ��h ) Petrov-Galerkin2.572471e+00 2.536110e+00 2.713026e+00 2.692954e+001.024982e+00 1.053929e+00 1.179669e+00 1.126765e+006.150985e-01 6.347660e-01 6.142472e-01 6.178719e-012.615208e-01 2.760115e-01 2.618566e-01 2.615416e-019.573677e-02 1.511278e-01 9.606547e-02 9.581336e-023.388482e-02 2.023391e-01 3.417012e-02 3.390462e-021.193008e-02 2.779688e-01 1.237064e-02 1.193274e-024.204846e-03 4.476061e-01 4.804128e-03 4.204771e-03Referenes1. Ben Belgaem F., Maday Y. (1994) A Spetral Element Methodology Tuned toParallel Implementations, CMAME, 116, 59-67.2. Ben Belgaem F., Maday Y. (1997) The mortar element method for three-dimensional �nite elements, M2AN, 31, 289-302.3. Bernardi C., Maday Y., Patera A. (1994) A new nononforming approah todomain deomposition: the mortar element method, in Nonlinear Partial Di�er-ential Equations and Their Appliations, H. Brezis and J.L. Lions, eds. Pitman,13{51.4. Braess D., Dahmen W. (1998) Stability estimates of the mortar �nite elementmethod for 3-dimensional problems, East-West J. Num. Math., 6, 249-263.5. Cazabeau L., Laour C., Maday Y. (1997) Numerial quadratures and mortarmethods, Computational Siene for the 21st Century, John Wiley and Sons,119{128.6. Lions J.L., Magenes E. (1972) Non-homogeneous Boundary Value Problems andAppliations I, Springer-Verlag, New York.7. Wohlmuth I.B. (2001) Disretization methods and iterative solvers based on do-main deomposition, Leture Notes in Computational Sienes and Engineering,17, Springer.