Universität Stuttgart

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1 Universität Stuttgart Second Order Lagrange Multiplier Spaces for Mortar Finite Elements in 3D Bishnu P. Lamichhane, Barbara I. Wohlmuth Berichte aus dem Institut für Angewandte Analysis und Numerische Simulation Preprint 2003/007

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3 Universität Stuttgart Second Order Lagrange Multiplier Spaces for Mortar Finite Elements in 3D Bishnu P. Lamichhane, Barbara I. Wohlmuth Berichte aus dem Institut für Angewandte Analysis und Numerische Simulation Preprint 2003/007

4 Institut für Angewandte Analysis und Numerische Simulation (IANS) Fakultät Mathematik und Physik Fachbereich Mathematik Pfaffenwaldring 57 D Stuttgart WWW: ISSN c Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors. IANS-Logo: Andreas Klimke. L A T E X-Style: Winfried Geis, Thomas Merkle.

5 Second Order Lagrange Multiplier Spaces for Mortar Finite Elements in 3D BishnuP.Lamichhane andbarbarai.wohlmuth Abstract Domain decomposition techniques provide a flexible tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements in 3D with different Lagrange multiplier spaces. In particular, we focus on Lagrange multiplier spaces which yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We present the discretization errors in different norms for linear and quadratic mortar finite elements with different Lagrange multiplier spaces. Numerical results illustrate the performance of our approach. Key words. mortar finite elements, Lagrange multiplier, dual space, domain decomposition, nonmatching triangulation. AMS subject classification. 65N30, 65N55. Introduction The coupling of different discretization schemes or of nonmatching triangulations can be analyzed within the framework of mortar methods. These nonconforming domain decomposition techniques provide a more flexible approach than standard conforming approaches. The nonconforming approach is of particular interest in many situations, for example, in problems with discontinuous diffusion coefficients and local anisotropies, when different parameters dominate different parts of the simulation domain or different discretization schemes are used in different subdomains. A complex global domain can be decomposed into several small subdomains of simple structure, and these subdomains can be meshed independently. To obtain a stable and optimal discretization scheme for the global problem, the information transfer among the subdomains has to be analyzed. Mortar methods were originally introduced to couple spectral and finite element approximations,see[bdm90,bmp93].anoptimalaprioriestimateinthe H -normforthemortar finite element method has been established in[bmp93, BMP94, Ben99, BM97]. The analysis of three-dimensional mortar finite elements is given in[bm97, KLPV0, BD98], and a hpversionisstudiedin[ss00].inthelinear3dcase,thestabilityofthemortarprojection for the standard Lagrange multiplier space is established in[bd98], where also a multigrid ThisworkwassupportedinpartbytheDeutscheForschungsgemeinschaft,SFB404,C2. 5

6 methodforthesaddlepointproblemisdiscussed.themainideaofthemortartechniqueis to replace the strong continuity condition of the solution across the interface by a weak one. Here,weconsidermortarmethodsforsecondorderfiniteelementsin3D.WefocusonLagrange multiplier spaces which yield locally supported basis functions of the constrained mortar space. Thepaperisorganizedasfollows:Intherestofthissection,wepresentourmodelproblem and briefly review the mortar method. In Section 2, we give some sufficient conditions on Lagrange multiplier spaces for quadratic finite elements and show the optimality of the approach. In Section 3, we present some examples of Lagrange multiplier spaces for quadratic finite elements for hexahedral triangulations. In contrast to earlier approaches, we use Lagrange multiplier spaces yielding a sparse inverse of the mass matrix. We also use the so-called dual Lagrange multiplier spaces which are biorthogonal to the trace of the finite element space at the interface. Unfortunately, a locally defined dual Lagrange multiplier space containing the bilinear hat functions does not exist for the serendipity elements.inthatcase,weaugmentthespacebyfacebubblefunctions.finallyinsection4,we present some numerical results in 3D for different Lagrange multiplier spaces illustrating the flexibility and performance of our approach. In particular, we consider the discretizationerrorsinthe L 2 -norm,theenergynormandinaweighted L 2 -normforthelagrange multiplier. We consider the following elliptic second order boundary value problem div(a u) + cu = f, in Ω, u = 0, on Ω, () where 0 < a 0 a L (Ω), f L 2 (Ω), 0 c L (Ω),and Ω Ê 3 isabounded polyhedral domain. The domain Ω is decomposed into K non-overlapping polyhedral subdomains Ω k, k =,, K,suchthat Ω = K Ω k with Ω i Ω j = for i j. k= Here, we consider only geometrically conforming situations where the intersection betweentheboundariesofanytwodifferentsubdomains Ω l Ω k, k l,iseitherempty, acommonedgeoraface. Wedefine Γ kl := Ω k Ω l, k, l K,theintersection of the boundaries of two subdomains and select only disjoint and non-empty interfaces γ k, k N.Moreover,each γ k canbeassociatedwithacouple k < k 2 Ksuch that γ k = Ω k Ω k2.oneachsubdomain,wedefine H (Ω k ) := {v H (Ω k ), v Ω Ωk = 0}, k =,, K and consider the unconstrained product space X := K H (Ω k). k= To impose a weak matching condition on the interface, a Lagrange multiplier space M := N k= H /2 (γ k )isintroducedontheskeleton Γ = N γ k. k= 6

7 Then,theweakmatchingconditionontheskeleton Γisrealizedintermsofthe L 2 -orthogonality of the jump [u] of the solution across the interface and the Lagrange multiplier space M [u] µ dσ = 0, µ M. Γ Eachsubdomain Ω k isassociatedwithashaperegularfamilyofhexahedraltriangulations T k;hk,themeshsizeofwhichisboundedby h k.wedenotethediscretespaceofconforming piecewisetriquadraticfiniteelementsorofserendipityelementson Ω k associatedwith T k;hk by X hk H (Ω k ).Now,theunconstrainedfiniteelementspace X h canbewrittenas X h := K X hk. k= Eachinterface γ k inheritsatwo-dimensionaltriangulation S k;hk eitherfrom T k;h k or T k2;h k2. The subdomain from which the interface inherits its triangulation is called slave or nonmortarside,theoppositeonemasterormortarside. Inthefollowing,wedenotethe indexoftheslavesideby s(k)andtheoneofthemastersideby m(k). Hence,theelementsof S k;hk areboundaryfacesof T s(k);hs(k) withameshsizeboundedby h s(k). Furthermore, we assume that the mesh on Γ is globally quasi-uniform, and each element in S k;hk, k =,, N,canbeaffinelymappedtothereferenceelement ˆT := (0, ) (0, ). ThediscreteLagrangemultiplierspace M h on Γisdefinedas M h := N k= M h(γ k ).Then, thediscreteweakmatchingconditionfor v h X h canbewrittenas [v h ] µ i dσ = 0, i n k, k N, (2) γ k where n k := dimm h (γ k )and {µ i } i nk formsabasisof M h (γ k ). Here, [v h ]isthejump ofthefunction v h on γ k fromthemastersidetotheslaveside. Asusual, s,ωk and (, ) s,ωk denotethenormandthecorrespondinginnerproducton H s (Ω k ),respectively, and s,ωk standsfortheseminorm.thenormon H /2 00 (γ k)anditsdualspace H /2 (γ k ) willbedenotedby /2 H 00 (γ k) and /2,γ k,respectively.wedefinethebrokennorm s on Xandthebrokendualnorm M on Mby K u 2 s := u 2 s,ω k, k= N and µ 2 M := µ 2 /2,γ k, respectively. Therearetwomainapproachestoobtainthemortarsolution u h X h of a discrete variational problem. The first one is based on the positive definite variational problemontheconstrainedfiniteelementspacewhichisgivenbymeansoftheglobal Lagrangemultiplierspace M h V h := {v h X h b(v h, µ h ) = 0, µ h M h }, where b(v h, µ h ) := N k= γ k [v h ] µ h dσ.weremarkthattheelementsofthespace V h satisfy a weak continuity condition on the skeleton Γ in terms of discrete Lagrange multiplier space M h.however, V h is,ingeneral,notasubspaceof H0 (Ω).Then,thevariationalfor- mulationofthemortarmethodcanbegivenintermsoftheconstrainedspace V h : find u h V h suchthat a(u h, v h ) = (f, v h ) 0, v h V h. (3) k= 7

8 Here,thebilinearform a(, )isdefinedas a(v, w) := K k= Ω k a v w + cv w dx. The second approach is based on enforcing the weak continuity condition on the skeleton Γasanadditionalvariationalequationwhichleadstoasaddlepointproblemonthe unconstrainedproductspace X h,see[ben99]:find (u h, λ h ) X h M h suchthat a(u h, v h )+ b(v h, λ h ) = (f, v h ) 0, v h X h, b(u h, µ h ) = 0, µ h M h. (4) ItisclearthatthechoiceofthediscreteLagrangemultiplierspace M h playsanessential role for the stability of the saddle point problem and the optimality of the discretization scheme. The Lagrange multiplier space has to be large enough to obtain an optimal consistencyerror,andithastobesmallenoughtogetanoptimalbestapproximationerroranda suitable discrete inf-sup condition. In the next section, we state sufficient conditions on the Lagrange multiplier space for quadratic finite elements to get optimal a priori estimates. Here, the nodal Lagrange multiplier basis functions are defined locally and are associated withtheinteriornodesofthemeshon γ k, k =,, N. Now,wegroupthedegreesof freedomof X h associatedwiththeskeleton Γintotwogroups u h Γ := (u m, u s ),where u m containsallnodalvaluesof u h onthemastersidesandallnodalvaluesatthenodesonthe boundaryoftheinterface γ k ontheslavesides,and u s consistsofallnodalvaluesof u h at theinteriornodesof γ k ontheslavesides, k N,seeFigure. Theassociatedsets ofnodesarecalled N m and N s,respectively.furthermore,wedenoteby N h thesetofall nodesin X h andweset N i := N h \(N m N s ).Thecorrespondingnodalvaluesof u h in N i willbedenotedbyablockvector u i.then(2)canbewritteninitsalgebraicformas M s u s + M m u m = 0. (5) Theentriesofthemassmatricesaregivenby m ij := γ k [φ j ] µ i dσ,where φ j arethefinite u m u s master side slave side Figure:Decompositioninto u m and u s fortheserendipityelements elementbasisfunctionscorrespondingtothedifferentgroupsofnodes,and µ i denotethe basisfunctionsof M h. Sincethebasisfunctionshavealocalsupport,themassmatrices aresparse. Formally,wecanobtainthevaluesontheslavesideas u s = Ms M m u m. Although M s isasparsematrix,theinversionof M s is,ingeneral,expensive,and Ms is 8

9 dense. This observation motivates our interest in Lagrange multiplier spaces which yield a sparseinverseofthemassmatrix M s.anaturalchoiceisaduallagrangemultiplierspace, see,e.g.,[woh0],havingadiagonalmassmatrix M s.then,thebasisfunctions {µ i } i nk of M h (γ k )and {ϕ i } i nk ofthetracespace W 0,h (γ k )havingthezeroboundarycondition on γ k satisfythebiorthogonalityrelation µ i ϕ j dσ = δ ij ϕ j dσ, i, j n k. (6) γ k γ k If M s isasparsetriangularmatrix,theinverseof M s isalsoasparsetriangularmatrix. Hence,togetasparseinverseofmassmatrix M s,itisenoughtoworkwithalagrange multiplierspacewhichyieldsasparsetriangularmassmatrix M s.wedefinetheproduct space W 0,h as N W 0,h := W 0,h (γ k ), k= andthebroken H /2 00 -normon W 0,hisdenotedby v 2 W := N k= v 2 H /2 00 (γ k). 2 A priori estimates In this section, we give some assumptions on quadratic Lagrange multiplier spaces which guarantee optimal a priori estimates. Following a similar approach as in[klpv0], we impose the following assumptions on the discrete Lagrange multiplier spaces for quadratic finite elements [P0] dimw 0,h (γ k ) = dimm h (γ k ), k N. [P] There is a constant C independent of the triangulation such that inf µ M h (γ k ) v µ 0,γ k Ch 2 s(k) v 2,γ k, v H 2 (γ k ), k N. [P2] There is a constant C independent of the triangulation such that θ 0,γk C sup µ M h (γ k )\{0} (θ, µ) 0,γk µ 0,γk, θ W 0,h (γ k ), k N. Itfollowsfromassumption[P]that P (γ k ) M h (γ k )forall k =,, N,where P (γ k ) isthespaceoflinearfunctionson γ k. Foreach γ k,themortarprojection Π k : L 2 (γ k ) W 0,h (γ k )isdefinedas Π k v µ dσ = γ k v µ dσ, µ M h (γ k ). γ k (7) The stability of the mortar projection is essential for the optimality of the best approximation error. 9

10 Lemma.Undertheassumptions[P0]and[P2],themortarprojection(7)isstableinthe L 2 -norm. Furthermore,if w H 0(γ k ) Π k w,γk C w,γk. Proof: Byassumption[P2],wefindthatif v W 0,h (γ k )satisfies (v, µ) 0,γk = 0forall µ M h (γ k ),then v = 0.Hence,themortarprojectioniswell-definedbytheassumptions [P0]and[P2].The L 2 -stabilityof Π k followsfrom Π k w 0,γk C sup µ M h (γ k )\{0} (Π k w, µ) 0,γk (w, µ) 0,γk = C sup C w 0,γk. µ 0,γk µ M h (γ k )\{0} µ 0,γk Now,usingthe L 2 -stabilityandaninverseestimate,wefindfor w H0 (γ k) ( ) Π k w,γk Π k w Pw,γk + Pw,γk C Π k (w Pw) 0,γk + w,γk h s(k) ( ) C w Pw 0,γk + w,γk C w,γk, h s(k) where Pdenotesthe L 2 -projectiononto W 0,h (γ k ). UsingLemmaandaninterpolationargument,weobtainfor w H /2 00 (γ k), Π k w H /2 00 (γ k) C w H /2 00 (γ k). Inanextstep,weprovidethebestapproximationpropertyofthespace V h. Weusethe ideas and techniques introduced in[bm97, BMP93]. Lemma2. Assumethattheassumptions[P0] [P2]hold.If u H 0 (Ω)and u Ω k H 3 (Ω k )for all k =,, K,thenthereexistsaconstant Cdependingonlyontheratioofthemeshsizesofthe master and slave sides such that inf u h V h u u h 2 C K h 4 k u 2 3,Ω k. k= Proof: Since W 0,h (γ k ) H /2 00 (γ k),each v W 0,h (γ k )cantriviallybeextendedtoafunction ṽ H /2 ( Ω s(k) ). Let H h ṽ H (Ω s(k) )bethediscreteharmonicextensionof ṽon Ω s(k). Then, H h ṽ,ωs(k) C ṽ H /2 ( Ω s(k) ) C v H /2 00 (γ k). Bymeansofthisdiscrete harmonicextension,wedefineadiscreteextensionoperator E k : W 0,h (γ k ) X h foreach γ k as E k v := H h ṽon Ω s(k),and E k v := 0elsewhere.Then E k v C v H /2 00 (γ k), v W 0,h(γ k ). (8) Let l h u X h bethelagrangeinterpolantof uin X h. Itiseasytoseethat v := l h u + N k= E kπ k [l h u]isanelementof V h.then,wefind N u v u l h u + E k Π k [l h u]. 0 k=

11 Byusing(8)andacoloringargument,wehave N K E k Π k [l h u] 2 = N N E k Π k [l h u] 2,Ω l C Π k [l h u] 2. H /2 00 (γ k) k= l= k= Wenotethattheconstant Cdoesnotdependonthenumberofsubdomains.Applyingthe L 2 (γ k )-stabilityof Π k andaninverseestimate,weget Π k [l h u] 2 H /2 00 (γ k) C h s(k) [l h u] 2 0,γ k k= k= k= C ( ) (u l h u) Ωm(k) 2 0,γ h k + (u l h u) Ωs(k) 2 0,γ k s(k) C ( ) h 5 m(k) h u 2 3,Ω m(k) + h 5 s(k) u 2 3,Ω s(k). s(k) Summingoverall k =,, N,weobtain ( ) N N N E k Π k [l h u] 2 C h mr h 4 m(k) u 2 3,Ω m(k) + h 4 s(k) u 2 3,Ω s(k), where { hm(k) h mr := max, h s(k) k= } k N. Finally,thelemmafollowsbyusingtheinterpolationpropertyof l h u. Weremarkthatincontrasttoaconvergencetheoryofmortarfiniteelementsin2D,the constantintherighthandsidedependsontheratioofthemeshsizesofmasterandslave sides.however,ifthemeshonthewirebasketisconformingand uiscontinuous,wefind [l h u] H /2 00 (γ k)andthusthe H /2 00 -stabilityof Π kcanbedirectlyapplied. Inthatcase, the ratio does not enter in the upper bound, see[klpv0]. Working with mesh dependent normsandatrivialextensionshowsthattheglobalratio h mr canbereplacedbyalocal one. Theorem3. Let uand u h bethesolutionsoftheproblems()and(3),respectively.assumethat u H0(Ω), u Ωk H 3 (Ω k )for k =,, K,and [a u n ] = 0on Γ. Undertheassumptions [P0] [P2], there exists a constant C depending only on the ratio of the meshsizes of the master and the slave sides such that K u u h 2 C h 4 k u 2 3,Ω k. Proof: Thebilinearform a(, )iscontinuouson X,anditiscoerciveon { } B := v v H (Ω k ), k K,and [v] dσ = 0, k N, γ k k= see[bmp93].hence,assumption[p]assuresthat V h B.Thus,Strang slemma[bs94] canbeapplied,andweget ( ) a(u u h, v h ) u u h C inf u v h + sup. (9) v h V h v h v h V h \{0}

12 Thefirsttermintherightsideof(9)denotesthebestapproximationerrorandthesecond one stands for the consistency error. Lemma 2 guarantees the required order for the best approximation error. Thus it is sufficient to consider the consistency error in more detail. Now, a(u u h, v h )canbewrittenas a(u u h, v h ) = Γ a u n [v h] dσ = N k= ( a u ), [v h ], v h V h. n k 0,γ k Here, u u nistheoutwardnormalderivativeof uon Γfromthemasterside,and n = u n k on γ k.wetake µ M h (γ k ),thenthedefinitionof V h yields ( a u ) (, [v h ] = a u ) µ, [v h ]. n k 0,γ k n k 0,γ k Since µ M h (γ k )isarbitrary,wecanboundtheconsistencyerrorby ( a u ) u, [v h ] inf a µ n k 0,γ µ M h (γ k ) k n (H /2 (γ k k )) [v h] /2,γk. Let Q : L 2 (γ k ) M h (γ k )bethe L 2 -projectiononto M h (γ k ).Usingassumption[P2]andan interpolationbetween L 2 (γ k )and H 2 (γ k ),weobtain w Qw 0,γk Ch 3/2 s(k) w 3/2,γ k, w H 3/2 (γ k ). In terms of a standard Aubin Nitsche trick and the previous estimate, we get w Qw (H /2 (γ k )) Ch/2 s(k) w Qw 0,γ k Ch 2 s(k) w 3/2,γ k. Finally, the trace theorem yields ( a u ), [v h ] Ch 2 u s(k) n k 0,γ k n k [v h ] /2,γk 3/2,γk Ch 2 s(k) u 3,Ω s(k) ( vh,ωm(k) + v h,ωs(k) ). Now,usingtheCauchy Schwarzinequalityandsummingoverall k =,, N,weobtain ( K ) /2 a(u u h, v h ) C v h h 4 k u 2 3,Ω k. k= To obtain an a priori estimate for the Lagrange multipliers, we follow exactly the same procedures as in[ben99]. Lemma4.AssumethattheLagrangemultiplierspace M h satisfiesassumptions[p0] [P2].Then, forevery µ M h,thereexistsav µ X h suchthat v µ C µ M, µ 2 M Cb(v µ, µ) and [v µ ] W C µ M. 2

13 Proof: Bymeansofthestabilityofthemortarprojection,wegetfor µ M h (γ k ) µ /2,γk = sup ϕ H /2 00 (γ k)\{0} = C sup ϕ W 0,h (γ k )\{0} (µ, ϕ) 0,γk ϕ /2 H 00 (γ k) (µ, ϕ) 0,γk ϕ H /2 00 (γ k) C sup ϕ H /2 00 (γ k)\{0} (µ, Π k ϕ) 0,γk Π k ϕ H /2 00 (γ k) C(µ, ϕ k ) 0,γk (0) forsome ϕ k W 0,h (γ k )with ϕ k H /2 00 (γ k) =.Now,weextend ϕ k W 0,h (γ k )to X h by usingtheextensionoperator E k asdefinedinlemma2toget E k ϕ k =: v k X h.then,we have v k C ϕ k H /2 00 (γ k) and 0 (µ, ϕ k ) 0,γk = b(v k, µ). Setting v µ := N k= b(v k, µ)v k andusingthefactthat v k C,weget N K N v µ 2 = b(v k, µ)v k 2 = b(v k, µ)v k 2,Ω l k= k= l= k= N N C b(v k, µ) 2 C µ 2 /2,γ k = C µ 2 M. This proves the first assertion. Using a coloring argument, the constant C does not depend on the number of subdomains. Furthermore, summing the equation(0) over all interfaces γ k, k =,, N,weobtainthesecondassertion µ 2 M C k= N b(v k, µ) 2 = Cb(v µ, µ). k= Finally, the third assertion follows from [v µ ] 2 W = N b(v k, µ) 2 [v k ] 2 = N b(v H /2 00 (γ k, µ) 2 k) k= k= N k= µ 2 /2,γ k [v k ] 2 H /2 00 (γ k) = µ 2 M. The best approximation property, Lemma 4, assumption[p2] and the first equation of the saddle point problem give an a priori estimate for the Lagrange multipliers. Lemma5.UndertheassumptionsofTheorem3,wehave λ λ h 2 M C K h 4 k u 2 3,Ω k. k= 3 Quadratic Lagrange multiplier spaces in 3D In this section, we consider different possibilities for Lagrange multiplier spaces in 3D for quadraticfiniteelementswithsuppϕ i =suppµ i.inparticular,wefocusonthestandard finite elements and the serendipity elements and restrict ourselves to hexahedral triangulations. These two finite elements have different degrees of freedom on the interface and therefore, the Lagrange multiplier spaces should be considered separately. 3

14 3. A dual Lagrange multiplier space for the standard triquadratic finite elements In the case of a hexahedral triangulation, a dual Lagrange multiplier space in 3D for trilinearandtriquadraticfiniteelementscanbeformedbytakingthetensorproductofthedual Lagrangemultiplierspacein2D.Let ˆϕ 0, ˆϕ and ˆϕ 2 bethenodalquadraticfiniteelement basisfunctionsonthereferenceelement (0, )inonedimension,where ˆϕ 0 and ˆϕ arethe basis functions corresponding to the left and the right vertices of the reference element, and ˆϕ 2 isthebasisfunctioncorrespondingtothemidpointofthereferenceelement.then,the quadratic dual Lagrange multiplier basis functions on the reference element are defined by ˆλ 0 (t) := ˆϕ 0 (t) 3 4 ˆϕ 2(t) + 2, ˆλ (t) := ˆϕ (t) 3 4 ˆϕ 2(t) + 2 and ˆλ 2 (t) := 5 2 ˆϕ 2(t). The Lagrange multiplier basis functions for the element touching a crosspoint should be modified.inparticular,if t = 0isacrosspoint,wehave andif t = isacrosspoint,weset ˆλ 2 (t) := 2t + 2, ˆλ (t) := 2t, ˆλ 2 (t) := 2t, ˆλ0 (t) := 2t. Furthermore,forahatfunction φ l pataninteriorvertex p,wefind φ l p(t) = µ p (t) + 2 (µ e(t) + µ e2 (t)), () where µ p isalagrangemultipliercorrespondingtothevertex pand µ e and µ e2 arethe basis functions associated with the midpoints of the two adjacent edges. If p is a crosspoint, wehave φ l p(t) = 2 µ e(t),where µ e isthelagrangemultiplierbasisfunctioncorresponding to the midpoint of the edge containing the crosspoint. Then, the Lagrange multiplier basis functionsonthereferenceelement ˆT = (0, ) (0, )havingatensorproductstructureare defined as ˆλ ij (x, y) := ˆλ i (x)ˆλ j (y). Here, ˆλ 00 (x, y), ˆλ 0 (x, y), ˆλ (x, y)and ˆλ 0 (x, y)arethelagrangemultiplierscorrespondingtothefourvertices (0, 0), (, 0), (, )and (0, ),and ˆλ 20 (x, y), ˆλ2 (x, y), ˆλ2 (x, y)and ˆλ 02 (x, y)aretheonescorrespondingtothemidpoints (0.5, 0), (, 0.5), (0.5, )and (0, 0.5) ofthefouredges,respectively,andfinally ˆλ 22 (x, y)istheonecorrespondingtothecenter of gravity (0.5, 0.5) of the reference element. The Lagrange multiplier basis functions are associated with the vertices, midpoints of the edges and the center of gravity of elements in S k;hk, k N.Theglobalbasisfunctions µ i areobtainedbyusinganaffinemapping andgluingthelocalonestogether.allnodesontheboundary γ k of γ k arecrosspointsand donotcarryadegreeoffreedomforthelagrangemultiplierspace.wenotethatwehave to use the modification at the crosspoints to compute the tensor product for the Lagrange multiplierscorrespondingtotheelementstouching γ k.observing(),wefindthatthe bilinearhatfunctionateachvertexiscontainedinthelagrangemultiplierspace M h (γ k ). Wepointoutthatthisisalsovalidon γ k,althoughtherearenodegreesoffreedom.hence, assumption[p] is satisfied. Assumption[P0] is trivially satisfied by construction. Now, 4

15 weverifyassumption[p2].let ϕ := n k k= a kϕ k bein W 0,hk (γ k )andset µ := n k k= a kµ k.in thefollowing,weassumethat ˆϕ i and ˆµ i areobtainedfrom ϕ i and µ i byanaffinemapping fromtheelement Ttothereferenceelement ˆT.Now,byusingthebiorthogonalityrelation (6) and the quasi-uniformity assumption, we get (ϕ, µ) 0,γk = n k i,j= n k a i a j (ϕ i, µ j ) 0,γk = i= a 2 i n k ϕ i dσ C a 2 i h2 s(k) C ϕ 2 0,γ k. γ k Takingintoaccountthefactthat ϕ 2 0,γ k µ 2 0,γ k n k i= a2 i h2 s(k),wefindthatassumption [P2] is satisfied. Figure 2 shows the three different types of Lagrange multipliers on the reference element. i= Figure 2: The Lagrange multipliers corresponding to a vertex(left), to an edge(middle) andtothecenterofgravity(left) 3.2 Serendipity elements and a dual Lagrange multiplier space Here, we give a non-existence result for a dual Lagrange multiplier space for the serendipity elements. A similar result for simplicial triangulations and quadratic finite elements is givenin[lw02].wedenoteby W h (γ k)thefiniteelementspaceofpiecewisebilinearhat functionson γ k. Incaseofstandardtriquadraticfiniteelements,thedualLagrangemultiplierspacewithtensorproductstructurecontains W h (γ k). Unfortunately,thereexists no dual Lagrange multiplier space for the serendipity elements yielding optimal a priori estimateswithsuppϕ i =suppµ i,where ϕ i aretheserendipitynodalfiniteelementbasis functionsontheinterface γ k. Lemma6.Undertheassumptionthatsuppϕ i =suppµ i,thereexistsnoduallagrangemultiplier space M h (γ k )suchthat W h (γ k) M h (γ k ). Proof: We prove this by contradiction. Assume that α i µ i = φ l p, (2) i 5

16 where φ l pisthebilinearhatfunctionassociatedwiththeinteriorvertex phavingthecoordinates (0, 0),seeFigure3.Supposethecoordinatesofthefourcornersoftheelement T be (, 0), (0, 0), (0, )and (, ),andoftheelement T 2 be (0, 0), (, 0), (, )and (0, ). ϕ j0 at (0, ) T T 2 φ l pat (0, 0) Figure 3: 2D interface of 3D hexahedral triangulation Becauseoftheduality,thefunctions µ i arebiorthogonaltothefiniteelementbasisfunctions ϕ i ontheinterface.hence,aftermultiplying(2)bysomefiniteelementbasisfunction ϕ j andintegratingovertheinterface γ k,weget α j = γ k ϕ j φ l p dσ γ k ϕ j dσ. Let j 0 betheinteriorvertexwithcoordinates (0, )suchthat j 0 and pshareoneedge,see Figure3.Then,wecanwrite ϕ j0 φ l p dσ = ϕ j0 φ l p dσ + ϕ j0 φ l p dσ = γ k T T 2 8 andthus α j0 0.Sincethebasisfunctions µ i arelocallylinearlyindependent,weobtain suppµ j0 supp i α iµ i.byconstruction,wefindsuppµ j0 suppφ l p,whichcontradicts (2). 3.3 Lagrange multiplier spaces for the serendipity elements The previous subsection shows that there does not exist a dual Lagrange multiplier space for the serendipity elements containing the bilinear hat function at each vertex and satisfyingsuppϕ i =suppµ i. Here,weconsidertwodifferentLagrangemultiplierspacesfor the serendipity elements. The essential point is that the Lagrange multiplier space should lead to an optimal and stable discretization scheme. For this purpose, the assumptions [P0] [P2] are crucial. The first idea is to choose a standard Lagrange multiplier space, see [BMP93,BMP94].Inthiscase,thebasisfunctionsforeachinteriorelement T S k;hk ofthe interface γ k (i.e., T S k;hk with T γ k = )aretheserendipityelementsin2d.allnodes on γ k donotcarryadegreeoffreedomforthelagrangemultipliers.therefore,inorder to satisfy assumption[p], it is necessary to modify the definition of the basis functions for theelementstouchingtheboundary γ k oftheinterface γ k.supposeanelement T S k;hk with T γ k has ndegreesoffreedomforthelagrangemultipliers. Then,thelocalLagrangemultiplierbasisfunction µ i atanode x i of Tischosentobeapolynomial 6

17 ofminimaldegreesuchthat µ i (x j ) = δ ij forall x j, j =,, n. Here, δ ij isthekronecker delta. These Lagrange multiplier basis functions are continuous. Since, in general, M h (γ k ) ( H /2 (γ k ) ),wecanworkwithdiscontinuouslagrangemultipliers. Working with a continuous Lagrange multiplier space which locally contains the linear functions has the advantage that assumption[p] is satisfied. Here too, assumption[p0] is trivially satisfiedbyconstruction.toverifyassumption[p2],wetake ϕ := n k k= a kϕ k in W 0,h (γ k ) anddefine µ := n k k= a kµ k.then (ϕ, µ) 0,γk = M ˆT = n k i,j= a i a j (ϕ i, µ j ) 0,γk = n k i,j= a i a j γ k ϕ i µ j dσ. Thelocalmassmatrix M ˆTforanelementhavingalltheverticesin γ k isgivenby Similarly, computing the local mass matrices for the different boundary cases, we find that alltheeigenvaluesofthelocalmassmatricesaregreaterthan 00 andsmallerthan 6.Then (ϕ, µ) 0,γk, ϕ 2 0,γ k and µ 2 0,γ k areequivalentto n k i= h2 s(k) a2 i,whichguaranteesassumption [P2].Thecouplingofthelocalmassmatricesyieldaglobalmassmatrixwhichissparsebut hasabandstructureofband-width O(/h).Thus,theinverseoftheglobalmassmatrix M s ontheslavesideisdense.asaconsequence,weobtainastiffnessmatrixassociatedwith the variational problem(3), which is not sparse. Then we cannot apply static condensation, and the multigrid method discussed in[wk0] cannot be used. To overcome this difficulty, we introduce a new approach for the serendipity elements. The ideaistousealagrangemultiplierspacewhichyieldsasparseinverseoftheglobalmass matrix M s ontheslaveside. Inthiscase,weusethetensorproductLagrangemultiplier spaceintroducedinsubsection3.. Tosatisfythecondition dimw 0,h (γ k ) = dimm h (γ k ), eachnon-emptyface f T Γoftheelement T oftheslavesideisenrichedwitha bubblefunction.thebubblefunction b H (T)correspondingtotheface fof Thasthe propertythat b T \f = 0and f b dσ 0.Wedefine K s := {T T s(k);hs(k), k N T Γcontainsatleastafaceof T },thesetofelementsontheslavesideeachofwhichhasat leastoneface fon Γ.Now,thespaceofbubblefunctions B h isformedby N s bubbles,where N s isthenumberofelementsin N k= S k;h k,andeachofthemiscorrespondingtoaface f ofanelement T K s,where f T Γ.Thisleadstooneadditionaldegreeoffreedom foreachnon-emptyface f T Γof T K s. Therearemanypossibilitiestodefine such a bubble function. Here, the triquadratic nodal finite element function associated withthecenterofgravityofthefaceisusedasabubblefunctioncorrespondingtothis face. Although we need only the restriction of bubble functions to the associated face to satisfy assumption[p0], each bubble function is supported on the whole element. Now, the

18 modifiedunconstrainedproductspace Xh tcanbewrittenas Xt h = Xs h B h,where Xh sis the unconstrained product space associated with the serendipity elements. In the sequel, thespace Xh twillbecalledaugmentedserendipityspaceandthecorrespondingelements augmentedserendipityelements.thisleadstoamassmatrix M s ontheslavesidehavinga specialstructure.suppose ˆϕ i and ˆλ i, ( i 9)bethelocalbasisfunctionsofthestandard triquadratic finite elements and their dual Lagrange multipliers, respectively. Here, the first four basis functions correspond to the vertices, the second four ones correspond to themidpointsoftheedges,andthelastonecorrespondstothecenterofgravityofthe referenceelement ˆT. Then,thelocalbasisfunctionsoftheserendipityelementscanbe writtenas ˆϕ s i = ˆϕ i + α i ˆϕ 9, ( i 8),where α i = 4 for ( i 4)and α i = 2 for (5 i 8).Usingthebiorthogonalityof ˆϕ i and ˆλ i,wehave ˆϕ s ˆλ i j dσ = (ˆϕ i + α i ˆϕ 9 ) ˆλ j dσ = δ ij ˆϕ i dσ + α i δ 9j ˆϕ 9 dσ. ˆT ˆT ˆT Infact,themassmatrixonthereferenceelement ˆTis M ˆT = (3) To show the consequence of our new Lagrange multiplier space, we consider the global massmatrix M s ontheslavesideinmoredetail.inthefollowing,weusethesamenotation forthevectorrepresentationofthesolutionandthesolutionasanelementin Xh tand M h. Thematrix Aisthestiffnessmatrixassociatedwiththebilinearform a(, )on Xh t Xt h,and thematrices Band B T areassociatedwiththebilinearform b(, )on Xh t M h.then,the algebraic formulation of the saddle point problem(4) is given by [ ] [ ] [ ] A B T uh fh =. (4) B 0 λ h 0 Werecallthegroupingofthedegreesoffreedomof Xh t introducedinsection. After augmentingtheserendipityspacewiththespaceofbubblefunctions B h,wefurtherdecomposethedegreesoffreedomassociatedwiththeinteriornodesof γ k, k N,on theslavesideintotwogroups (u s, u b ).Here,theblockvector u s containsnodalvaluesof uattheinteriornodesof γ k, k N,correspondingtotheverticesandedgesonthe slaveside,and u b standsforallnodalvaluescorrespondingtothebubblefunctionsonthe slaveside.withthisdecomposition,wecanwrite u T h = (ut i, ut m, u T s, u T b ).Theblockvector λ h containingthenodalvaluesofthelagrangemultiplierissimilarlydecomposedwith λ T h = (λt s, λt b ). Intermsofthisdecomposition,wecanrewritethealgebraicformofthe ˆT 8

19 saddle point problem(4) as A ii A im A is A ib 0 0 A mi A mm A ms A mb Mm T MT bm A si A sm A ss A sb D s Mbs T A bi A bm A bs A bb 0 D b 0 M m D s M bm M bs D b 0 0 u i u m u s u b λ s λ b = f i f m f s f b 0 0. (5) Recallingthealgebraicstructure(5)ofthebilinearform b(, )restrictedto Xh t M h,we have [ ] Mm D s 0 B =, M bm M bs D b where D b and D s arediagonalmatrices,and M bs, M m and M bm arerectangularmatrices. Thematrix D b isdiagonalduetothefactthatthebubblefunctionsaresupportedonly inoneelement,andthediagonalformof D s followsfromthestructureofthelocalmass matrix,see(3).hence,theglobalmassmatrix M s ontheslavesidecanbewrittenas [ ] Ds 0 M s =. M bs ThegreatbenefitofthisLagrangemultiplierspaceisthattheinverseofthemassmatrix M s canbecomputedveryeasily,andtheinverseissparse.infact,theinverseofthemass matrix M s is [ ] M s = D s D b D b M bs Ds 0 D b Thus,thesolutionattheslavesidedependslocallyonthesolutionatthemasterside.Here, wehavetoinvertonlytwodiagonalmatricesandscale M bs tocomputetheinverseofthe massmatrix M s.thestiffnessmatrixassociatedwiththevariationalproblem(3)issparse, and efficient iterative solver like multigrid can easily be adapted to the nonconforming situation. Additionally, the condition number of the mass matrix is better compared to the standard Lagrange multiplier space. Furthermore, the degree of freedom corresponding to thebubblefunctionscanlocallybeeliminatedbystaticcondensation.sincethematrix D b isdiagonal,thesixthandthefourthlineofthesystem(5)give u b = D b (M bm u m + M bs u s ), and λ b = D [ b fb A bi u i (A bm A bb D b M bm )u m (A bs A bb D ] b M bs )u s. Now,weeliminate u b and λ b fromthesystem(5)andobtainanewsystem Âû h = ˆF h, where û T h = (ut i, ut m, ut s, λt s ).Defining M := D b M bm and M 2 := D b M bs wehave A ii A im A ib M A is A ib M 2 0 Â = A mi M T A bi A mm A mb M M T (A bm A bb M ) A ms A mb M 2 M T (A bs A bb M 2) Mm T A si M2 T A bi A sm A sb M M2 T (A bm A bb M ) A ss A sb M 2 M2 T (A bs A bb M 2) D s, 0 M m D s 0. 9

20 andtherighthandsidecanbewrittenas ˆF h = f i f m M T f b f s M T 2 f b 0 Weobservethatthematrix Âissymmetric,if Aissymmetricandithasexactlythesame structure as the saddle point matrix arising from mortar finite element method with a dual Lagrange multiplier space, see[woh0]. Because of this structure of the algebraic system, we can apply the multigrid method proposed in[wk0]. Remark7.Thereisalsoapossibilitytousewaveletstogetamassmatrixofspecialstructuresothat the inversion can be cheaper, and the inverse is sparse. In[Ste00], locally supported and piecewise polynomial wavelets are studied on non-uniform meshes which give a lower triangular mass matrix with higher order finite elements in triangular meshes.. 4 Numericalresults Here, we present some numerical examples in 3D for linear and quadratic mortar finite elements. We consider three different cases for quadratic mortar finite elements. The first one is the standard triquadratic finite element space with the dual Lagrange multiplier space introduced in Subsection 3.. The second one is the serendipity space with a standard Lagrange multiplier space given in Subsection 3.2. Finally, the third one is the augmented serendipity space associated with the tensor product Lagrange multiplier space. Our numerical results show the same asymptotic behaviour as predicted by the theory. Theimplementationisbasedonthefiniteelementtoolboxug,[BBJ + 97]. Wedonotdiscuss and analyze an iterative solver for the arising linear systems. Working with dual Lagrange multiplier spaces has the advantage that the flux can locally be eliminated, and static condensation yields a positive definite system on the unconstrained product space. In[WK0,Woh0],themodificationofthesystemhasbeencarriedoutandalocalmodification of the transfer operators of lower complexity has been proposed. The introduced multigrid has a level-independent convergence rate and is of optimal complexity. Unfortunately, in the case of a standard Lagrange multiplier space no local elimination of the flux can be carried out. Following the approach in[wk0], the sparsity of the modified systemandtheefficiencyofthemultigridsolverislost.inthatcase,weapplyamultigrid method for saddle point problems. This technique has been considered for mortar elements in[ww98]andfurtheranalyzedin[bdw99,ww02].itturnsoutthatwedonothaveto workinapositivedefinitesubspace,andthesmoothercanberealizedinaninnerandouter iteration scheme. As in the other approach, level-independent multigrid convergence rates can be established. However, the numerical solution process is slower if we have to work with the saddle point approach. We point out that the more efficient multigrid method for themodifiedpositivedefinitesystemcanonlybeappliedwhentheinverseof M s issparse, whereasthesaddlepointmultigridmethodismoregeneral.althoughwedonothavea dual Lagrange multiplier space for the serendipity elements, the tensor product Lagrange multiplier space in combination with the introduction of bubble functions on the slave side of the interface yield an optimal discretization scheme. We present some numerical results in 3D illustrating the performance of the different Lagrange multiplier spaces. In particular, 20

21 wecomparethediscretizationerrorsinthe L 2 -and H -normforthesolutionforlinearand quadratic mortar finite elements. The discretization errors in the flux across the interface are compared in a mesh-dependent Lagrange multiplier norm, which is defined by µ µ h 2 h := N m= T S m;hm h T µ µ h 2 0,T, where h T isthediameteroftheelement T.Forallourexamples,wehaveuseduniformrefinement. In each refinement step, one element is refined into eight subelements. We denote by X l h and Xf h theunconstrainedfiniteelementspacesassociatedwiththestandardfinite elementspacesforthetrilinearandthetriquadraticcase,respectively. Similarly, X s h and X t h aretheunconstrainedfiniteelementspacesassociatedwiththeserendipityelements and the augmented serendipity elements as defined in the previous section, respectively. Thecorrespondingsolutionsaredenotedby u l h, uf h, us h and ut h,respectively. Remark 8. We note that the concept of dual Lagrange multiplier spaces can be generalized to distorted hexahedral meshes. In that case, the mapping between the actual element and the reference elementhasanon-constantjacobian. Asaconsequence,wehavetocomputeforeachfaceonthe interface a biorthogonal basis with respect to local nodal one. This can be easily done by solving a local mass matrix system. By construction, the sum of the local dual Lagrange multiplier basis functions is one. Defining the global Lagrange multiplier basis functions by gluing the local ones together, we find that the constants are included in the Lagrange multiplier space. As a consequence, itiseasytoverifythatthediscretizationerrorisoforder hforthelowestorderconformingfinite elements. Ourfirstexampleisgivenby u = f on Ω := (0, ) 2 (0, 2),where Ωisdecomposedintotwosubdomains Ω := (0, ) 3 and Ω 2 := (0, ) 2 (, 2). Therighthand side fandtheboundaryconditionsarechosensuchthattheexactsolutionisgivenby u(x, y, z) = 2 exp( x 2 y 2 z 2 )sin(2 yx)(x + y + z)(x y z).infigure4,thedecomposition of the domain, the initial triangulation and the isolines of the solution at the interface z = are shown. We have used a nonconforming initial triangulation, where the Figure 4: Decomposition of the domain and initial triangulation(left), isolines of the solution at the interface(right), Example lowercubeisrefinedonce,andtheuppercubeisleftasitis.theerrorsinthe L 2 -, H -and the weighted Lagrange multiplier norms are given in Tables 3. The tables show that we getthepredictedasymptoticconvergencerates.forthequadraticcase,theerrorsinthe L 2-2

22 and H -normareoforder h 3 and h 2,whereastheyareoforder h 2 and hforthelinearcase. The errors in the weighted Lagrange multiplier norm for the quadratic and linear cases areoforder h 5/2 and h 3/2,respectively.Theoretically,theerrorsintheweightedLagrange multipliernormforquadraticandlinearcaseareexpectedtobeoforder h 2 and h,respectively. Thebetterconvergenceratesareduetothefactthattheerrorinthe H -normis equallydistributedandthelagrangemultiplierspacehasan O(h 5/2 )and O(h 3/2 )approximation property in the considered norm. For the three different second order approaches, thequantitativeresultsareasymptoticallyalmostthesameinthe L 2 -and H -norm.only in the weighted Lagrange multiplier norm, the full triquadratic yields better results. Table:Discretizationerrorsinthe L 2 -norm,(example) level #elem. u u l h 0 u u f h 0 u u s h 0 u u t h e e-0 9.2e e e e e e e e e e e e e e e e e e-04 Table2:Discretizationerrorsinthe H -norm,(example) level #elem. u u l h u u f h u u s h u u t h e e e e e e e e e e e e e e e e e e e e-02 Table 3: Discretization errors in the weighted Lagrange multiplier norm,(example ) level #elem. λ λ l h h λ λ f h h λ λ s h h λ λ t h h e e e e e e e e e e e e e e e e e e e e-02 InExample2,wechooseaL-shapeddomain.Thedomain Ω := ((0, ) 2 (0, 2)) ([, 2) (0, ) 2 )isdecomposedintothreecubes, Ω := (0, ) 3, Ω 2 := (0, ) 2 (, 2)and Ω 3 := (, 2) (0, ) 2.Wehaveshownthedecompositionofthedomainandtheinitialtriangulationinthe leftpictureoffigure5,andtheisolinesofthesolutionattheinterface z = areshownin theright.asbefore,wesolveapoissonproblem u = f 22

23 with the right hand side function f and the Dirichlet boundary conditions determined by ( theexactsolution u(x, y, z) = (x ) 2 + (z ) 2) 5/6 ( cos 6 y 2 + x ). Figure 5: Decomposition of the domain and initial triangulation(left), isolines of the solutionattheinterface z = (right),example2 WehavetabulatedtheerrorsindifferentnormsinTables4 6. Here,thesolutionisnot H 3 -regular.therefore,wecannotexpectthesameorderofconvergenceasintheprevious example.sincethesolution u H 8/3 ǫ (Ω)for ǫ > 0,weexpectthesameorderofconvergence as in the previous example in the linear case. In the quadratic case, theoretically the ordersof h 8/3 andof h 5/3 areexpectedinthe L 2 -and H -norm,respectively.inallthree casesofquadraticfiniteelements,weobservetheasymptoticratesinthe L 2 -and H -norm, which are better than predicted by the theory. The quantitative results are also almost the sameinthesenorms. Heretoo,weobservethebetterconvergenceratesfortheerrorsin the weighted Lagrange multiplier norm. Table4:Discretizationerrorsinthe L 2 -norm,(example2) level #elem. u u l h 0 u u f h 0 u u s h 0 u u t h e e e e e e e e e e e e e e e e e e e e-04 InExamplesand2,thereisnotanysignificantdifferenceintheaccuracybetweenthedifferentquadraticmortarsolutionseitherinthe L 2 -normorinthe H -norm.however,there is some quantitative difference in the errors in the weighted Lagrange multiplier norm between different quadratic mortar solutions. In this norm, the standard triquadratic finite elements with the tensor product Lagrange multiplier space gives the best result, whereas the difference between the augmented serendipity elements with the tensor product Lagrange multiplier space and the serendipity elements with the standard Lagrange multiplier space is quite negligible in Example 2. In Example, the serendipity elements with the standard Lagrange multiplier space yields better result than the augmented serendipity elements with the tensor product Lagrange multiplier space. However, the difference is not 23

24 Table5:Discretizationerrorsinthe H -norm,(example2) level #elem. u u l h u u f h u u s h u u t h e e e e e e e e e e e e e e e e e e e e-02 Table 6: Discretization errors in the weighted Lagrange multiplier norm,(example 2) level #elem. λ λ l h h λ λ f h h λ λ s h h λ λ t h h e e e e e e e e e e e e e e e e e e e e-02 essential, and the asymptotic convergence orders are the same. We remark that for both of theseexamplestheasymptoticphasestartsfromthethirdstep.thisisduetothefactthat the initial triangulation is very coarse. For the next two examples, we consider only the serendipity elements. In our third example,thedomain Ω := (0, ) 2 (0, 2.5)isdecomposedintothreesubdomains Ω := (0, ) 3, Ω 2 := (0, ) 2 (, 2),and Ω 3 := (0, ) 2 (2, 2.5). Therighthandside fandtheboundaryconditionsof u = farechosensuchthattheexactsolutionisgivenby u(x, y, z) = 5(z.4)((x 0.5) 2 + 4(y 0.3) 3 ) + z(z )sin(4πxy)(2(x y) 2 + (y + x ) 2 ).InFigure 6, we have shown the decomposition of the domain, the initial nonmatching triangulation andtheisolinesofthesolutionattheinterface z = 2. Figure 6: Decomposition of the domain and initial triangulation(left), isolines of the solutionattheinterface z = 2(right),Example3 Here,wehavethreesubdomainsandtwointerfaces.Themiddlecubeistakenastheslave 24

25 side. We start with a nonconforming coarse initial triangulation having 23 elements. The errors along with their order of convergence at every step of refinement in different norms aregivenintables7 9.Asbefore,wegetthecorrectasymptoticratesforbothcasesofthe serendipityelements,andtheerrorsinthe L 2 -and H -normarealmostthesameforboth approaches. In the weighted Lagrange multiplier norm the serendipity elements yield less errors than the augmented serendipity elements. However, the difference is quite negligible, and the asymptotic rate of convergence is optimal in both cases. Table7:Discretizationerrorsinthe L 2 -norm,(example3) level #elem. u u l h 0 u u s h 0 u u t h e e e e e e e e e e e e e e e Table8:Discretizationerrorsinthe H -norm,(example3) level #elem. u u l h u u s h u u t h e e e e e e e e e e e e e e e Table 9: Discretization errors in the weighted Lagrange multiplier norm,(example 3) level #elem. λ λ l h h λ λ s h h λ λ t h h e e e e e e e e e e e e e e e Inourlastexample,weconsideradomain Ω := (0, 2) (0, ) (0, 2),whichisdecomposed intofoursubdomains Ω := (0, ) 3, Ω 2 := (0, ) 2 (, 2), Ω 3 := (, 2) (0, ) 2 and Ω 4 := (, 2) (0, ) (, 2). Wehaveshownthedecompositionofthedomainandtheinitial triangulationintheleftpictureoffigure7,theisolinesofthesolutionontheplane y = 2 inthemiddle,andthefluxoftheexactsolutionattheinterface x = isshownintheright one.here, Ω 2 and Ω 3 aretakentobetheslavesidesandtherestarethemastersides.the problem for this example is given by a reaction-diffusion equation div(a u) + u = f in Ω, 25

26 where aischosentobe in Ω and Ω 4,and a = 0in Ω 2 and Ω 3.Wehavechosentheexact solution u(x, y, z) = (x )y (z )exp( (x ) 2 y 2 (z ) 2 )cos(2 x + 2 y + 2 z)/a and the right hand side f and the Dirichlet boundary conditions are determined from the exact solution. We remark that the exact solution u has a jump in the normal derivative Figure 7: Decomposition of the domain and initial triangulation(left), isolines of the solutionattheplane y = 2 (middle)andfluxoftheexactsolutionattheinterface x = (right), Example 4 across the interface, whereas the flux is continuous. We have given the errors together with their order of convergence in every step of refinement in different norms in Tables 0 2.Asinotherexamples,wegetthesameasymptoticratesforthe L 2 -and H -normand better convergence rates in the weighted Lagrange multiplier norm. In contrast to other examples, we see that the asymptotic of errors in the weighted Lagrange multiplier norm forthequadraticcaseisevenbetterthan O(h 3 ). Table0:Discretizationerrorsinthe L 2 -norm,(example4) level #elem. u u l h 0 u u s h 0 u u t h e e e e e e e e e e e e e e e Table:Discretizationerrorsinthe H -norm,(example4) level #elem. u u l h u u s h u u t h e e e e e e e e e e e e e e e InbothExamples 3and 4,thediscretizationerrorsinthe L 2 -and H -normarealmostthe same for different approaches. Concerning the discretization errors in the weighted Lagrange multiplier norm, although we see some quantitative difference in Examples 3 and 26

27 Table 2: Discretization errors in the weighted Lagrange multiplier norm,(example 4) level #elem. λ λ l h h λ λ s h h λ λ t h h e e e e e e e e e e e e e e e , we observe the better convergence rate for both approaches. In this norm, the serendipity elements with the standard Lagrange multiplier space gives better result in both examples. However, in Example 3, the order of convergence is almost the same for both approaches, whereas the augmented serendipity elements show the better order of convergence in Example 4. We already observe the better rate of convergence from the augmented serendipity elements from the second step of refinement. In all our examples, we observe the optimal asymptotic convergence rates as predicted by the theory. Although we see the same qualitative behaviour, some quantitative differences can be observed. However, there is not any essential difference in the discretization errors between different quadratic mortar solutions. Because of the addition of bubble functions attheskeleton Γ, X t h hasmoredegreeoffreedomthan Xs h.however,thesebubblefunctions can locally be eliminated from the algebraic formulation of the saddle point problem leadingtoasystemmatrix,whichissimilartothealgebraicformofthesaddlepointproblem arising from the mortar discretization with a dual Lagrange multiplier space. On the otherhand,thegrowthrateofthenumberofbubblefunctionsisonlyafactoroffourin each refinement step, and restricted to the skeleton. This is negligible since we can work with the efficient multigrid solver in case of the augmented serendipity space with the tensor product Lagrange multiplier space. Although we can work with the efficient multigrid solver in case of standard triquadratic finite elements, the approach is not as optimal as the augmented serendipity approach due to the more degree of freedom and the growth rate offactoreightineachrefinementstep.itturnsoutthatthemostefficientapproachisthe one given by the augmented serendipity elements. The discretization errors are as good as in the other cases, and the numerical solution is cheaper. References [BBJ + 97] P.Bastian,K.Birken,K.Johannsen,S.Lang,N.Neuß,H.Rentz Reichert,and C. Wieners. UG a flexible software toolbox for solving partial differential equations. Computing and Visualization in Science, :27 40, 997. [BD98] D. Braess and W. Dahmen. Stability estimates of the mortar finite element method for 3 dimensional problems. East West J. Numer. Math., 6: , 998. [BDM90] C. Bernardi, N. Debit, and Y. Maday. Coupling finite element and spectral methods: First results. Math. Comp., 54:2 39, 990. [BDW99] D. Braess, W. Dahmen, and C. Wieners. A multigrid algorithm for the mortar finite element method. SIAM J. Numer. Anal., 37:48 69,