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Domain Decomposition Methods in Science and Engineering
Inthispaper,wedenoteby V areflexiveBanachspaceand K V isanonempty closedconvexsubset.Also, F W K ! R isaGâteauxdifferentiablefunctionaland weassumethatthereexisttworealnumbers p; q >1 suchthatforany M >0 there exist ˛M ;ˇM >0 forwhich
forany u;v 2 K , jjujj; jjv jj M .Inviewoftheseproperties,wecanprovethat F is aconvexfunctionaland 1< q 2 p.
Weconsiderthevariationalinequality u 2 K : < F 0 .u/;v u > 0; forany v 2 K ; (1) andif K isnotbounded,wesupposethat F iscoercive,i.e. F .v/ !1 as jjv jj! 1.Then,problem(1)hasanuniquesolution.Let V1 ; ; Vm besomeclosed subspacesof V forwhichwemakethefollowing.
Assumption1 ThereexistsaconstantC0 >0 suchthatforanyw;v 2 Kand wi 2 Vi withw C Pi jD1 wj 2 K,i D 1; ; m,thereexist vi 2 Vi ,i D 1; ; m, satisfying w C
One-levelmethodsareobtainedfromAlgorithm 1 byusingthefiniteelement spaces.Tothisend,weconsiderasimplicialregularmeshpartition Th ofmeshsize h over ˝ Rd .Also,let ˝ D[m iD1 ˝i beadomaindecompositionof ˝ ,the overlappingparameterbeing ı ,andweassumethat Th suppliesameshpartition foreachsubdomain ˝i , i D 1;:::; m.In ˝ ,weusethelinearfiniteelementspace Vh whosefunctionsvanishontheboundaryof ˝ and,foreach i D 1;:::; m,we considerthelinear finiteelementspace V i h Vh whosefunctionsvanishoutside ˝i Spaces Vh and V i h , i D 1;:::; m,areconsideredassubspacesof W 1; , 1 1, andlet Kh Vh beaconvexsetsatisfying.
Property1 If v; w 2 Kh ,andif 2 C 0 .˝/, j 2 C 1 . / forany 2 Th ,and 0 1,then Lh . v C .1 /w/ 2 Kh ; where Lh isthe P1 -Lagrangianinterpolation. Weseethattheconvexsetsofobstacletypesatisfythisproperty,andwehave(see Proposition3.1inBadea(2006)fortheproof)
Proposition1 Assumption 1 holdsforthelinearfiniteelementspaces,V D Vh and Vi D V i h ,i D 1;:::; m,andforanyconvexsetK D Kh Vh havingProperty 1.The constantC0 inAssumption 1 canbewrittenasC0 D C .m C 1/.1 C m 1 ı /; whereC isindependentofthemeshparameterandthedomaindecomposition.
Inthecaseofthetwo-levelmethods,weconsidertworegularsimplicialmesh partitions Th and TH on ˝ Rd , Th beingarefinementof TH .Besidesthefinite elementspaces Vh , V i h , i D 1;:::; m andtheconvexset Kh ,definedfortheonelevelmethods,weintroducethelinearfiniteelementspace V 0 H correspondingtothe H -level,whosefunctionsvanishontheboundaryof ˝ .Thetwo-levelmethodis obtainedfromthegeneralsubspacecorrectionAlgorithm 1 for V D Vh , K D Kh , andthesubspaces V0 D V 0 H , V1 D V 1 h , V2 D V 2 h , ::: , Vm D V m h .Also,thesespaces areconsideredassubspacesof W 1; , 1 1,andwehavethefollowing(see Proposition4.1inBadea(2006)fortheproof)
Proposition2 Assumption 1 issatisfiedforthelinearfiniteelementspacesV D Vh andV0 D V 0 H ,Vi D V i h ,i D 1;:::; m,andanyconvexsetK D Kh havingProperty 1. TheconstantC0 canbetakenoftheformC0 D Cm 1 C .m 1/ H ı Cd ; .H ; h/; whereCisindependentofthemeshanddomaindecompositionparameters,and
where ˝ R2 , K D Œa; b , a b, a; b 2 W 1; 0 .˝/, 1< < 1.Thesenumerical experimentshaveconfirmedtheprevioustheoreticalresults.
3MultilevelandMultigridMethods
DetailsconcerningtheresultsinthissectioncanbefoundinBadea(2014, 2015). Asinthecaseoftheone-andtwo-levelmethods,weconsiderproblem( 1).Let Vj , j D 1;:::; J ,beclosedsubspacesof V D VJ whichwillbeassociatedwiththe leveldiscretizations,and Vji , i D 1;:::; Ij ,beclosedsubspacesof Vj whichwillbe associatedwiththedomaindecompositionsonthelevels.Weconsider K V anon emptyclosedconvexsubsetandwrite I D max jDJ ;:::;1 Ij .
Togetsharpererrorestimationsinthecaseofthemultigridmethod,weconsider someconstants 0<ˇjk 1, ˇjk D ˇkj , j; k D J ;:::;1,forwhich hF 0 .v C vji / F 0 .v/;vkl i ˇM ˇjk jjvji jjq 1 jjvkl jj; forany v 2 V , vji 2 Vji , vkl 2 Vkl with jjv jj, jjv C vji jj, jjvkl jj M , i D 1;:::; Ij and l D 1;:::; Il .Also,wefix aconstant p p qC1 p andassumethatthereexistsaconstant C1 suchthat jj PJ jD1 PIj iD1 wji jj C1 .PJ jD1 PIj iD1 jjwji jj / 1 ; forany wji 2 Vji , j D J ;:::;1, i D 1;:::; Ij .Evidently,ingeneral,wecantake ˇjk D 1; j; k D J ;:::;1 and C1 D .IJ / 1 : Inthemultigridmethods,theconvexsetswherewelookforthecorrections areiterativelyconstructedfromaleveltoanotherduringtheiterationsinfunction ofthecurrentapproximation.Inthisgeneralbackgroundwemakethefollowing.
-atlevelJ:weassumethat 0 2 KJ ; KJ fvJ 2 VJ :w C vJ 2 K g andconsider awJ 2 KJ , -atalevelJ 1 j 1:weassumethat 0 2 Kj ; Kj fvj 2 Vj :w C wJ C ::: C wjC1 C vj 2 K g andconsiderawj 2 Kj . Also,wemakeasimilarassumptionwiththatinthecaseofthe-oneandtwo-level methods,
Assumption3 ThereexiststwoconstantsC2 ; C3 >0 suchthatforanyw 2 K, wji 2 Vji ,wj1 C ::: C wji 2 Kj ,j D J ;:::;1,i D 1;:::; Ij ,andu 2 K,thereexist uji 2 Vji ,j D J ;:::;1,i D 1;:::; Ij ,whichsatisfy uj1 2 Kj andwj1 C ::: C wji 1 C uji 2 Kj ; i D 2;:::; Ij ; j D J ;:::;1; u w D J X jD1 Ij X iD1 uji ; J X jD1 Ij X iD1 jjuji jj C2 jju wjj C C3 J X jD1 Ij X iD1 jjwji jj
Theconvexsets Kj ,j D J ;:::;1,areconstructedasinAssumption 2 withtheabove
Theorem2 UndertheaboveconditionsonthespacesandthefunctionalF,if Assumptions 2 and 3 hold,thenthereexistsanM >0 suchthat jjun jj M,for anyn 0,andwehavethefollowingerrorestimations: (i)ifp D q D 2 wehave jjun
.
/ ; (ii)ifp > qwehave jju un jjp p
TogetthemultilevelmethodcorrespondingtoAlgorithm 2,weconsiderafamily ofregularmeshes Thj ofmeshsizes hj , j D 1;:::; J ,overthedomain ˝ Rd and assumethat ThjC1 isarefinementof Thj .Let,ateachlevel j D 1;:::; J , f˝ i j g1 i Ij beanoverlappingdecompositionof ˝ ,ofoverlappingsize ıj .Wealsoassumethat, for 1 i Ij ,themeshpartition Thj of ˝ suppliesameshpartitionforeach ˝ i j , diam.˝ i jC1 / Chj and I1 D 1.
Weintroducethelinearfiniteelementspaces, Vhj Dfv 2 C .˝j / : v j 2 P1 . /; 2 Thj ;v D 0 on @˝j g, j D 1;:::; J ,correspondingtothelevelmeshes, and V i hj Dfv 2 Vhj : v D 0 in ˝j n˝ i j g, i D 1;:::; Ij ,associatedwiththelevel decompositions.Spaces Vhj j D 1;:::; J 1,willbeconsideredassubspacesof W 1; , 1 1.
Themultilevelandmultigridmethods willbeobtainedfromAlgorithm 2 fora twosidedobstacleproblem(1),i.e.theconvexsetisoftheform K Dfv 2 VhJ : ' v g; with '; 2 VhJ , ' .Concerningtheconstructionofthelevelconvex sets,wehave[Proposition3.1inBadea(2014)]
Proposition3 Assumption 2 holdsfortheconvexsets Kj ,j D J ;:::;1,defined as,
-forw 2 K,atthelevelJ,wetake 'J D ' w; J D w; KJ D Œ'J ; J ; andconsideranwJ 2 KJ ; -atalevelj D J 1;:::;1,wedefine 'j D Ihj .'jC1 wjC1 /; j D Ihj . jC1 wjC1 /; Kj D Œ'j ; j ; andconsideranwj 2 Kj ,Ihj :VhjC1 ! Vhj ; j D 1;:::; J 1; beingsomenonlinearinterpolationoperatorsbetweentwoconsecutivelevels. Also,oursecondassumptionholds[seeProposition2inBadea(2015)],
Proposition4 Assumption 3 holdsfortheconvexsets Kj ,j D J ;:::;1,definedin
Proposition 3.TheconstantsC2 andC3 arewrittenas
WeprovedthatAssumptions 2 and 3 hold,andhaveexplicitlywrittenconstants C2 and C3 infunctionofthemeshandoverlappingparameters.Wecanthenconclude fromTheorem 2 thatAlgorithm 2 isgloballyconvergent.Convergenceratesgivenin Theorem 2 dependonthefunctional F ,themaximumnumberofthesubdomainson eachlevel, I ,andthenumberoflevels J .Sincethenumberofsubdomainsonlevels canbeassociatedwiththenumberofcolorsneededtomarkthesubdomainssuch thatthesubdomainswiththesamecolordonotintersectwitheachother,wecan concludethattheconvergencerateessentiallydependsonthenumberoflevels J .
Inthegeneralframeworkofmultilevelmethodswetake C1 D CJ 1 maxk D1; ;J PJ jD1 ˇkj D J and,asfunctionsdependingonlyof J ,wehave
Intheabovemultilevelmethodsameshistherefinementofthatoneonthe previouslevel,butthedomaindecompositionsarealmostindependentfromone leveltoanother.Weobtainsimilarmultigridmethodsbydecomposingthedomain bythesupportsofthenodalbasisfunctionsofeachlevel.Consequently,the subspaces V i hj , i D 1;:::; Ij ,areone-dimensionalspacesgeneratedbythenodal basisfunctionsassociatedwiththenodesof Thj , j D J ;:::;1.Inthecaseof themultigridmethods,wecantake C1 D C andmaxk D1; ;J PJ jD1 ˇkj D C Nowwecanwritetheconvergencerate ofthemultigridmethodcorrespondingto Algorithm 2 infunctionofthenumberoflevels J foragivenparticularproblem.In Badea(2014),theconvergencerateofthemultigridmethodfortheexamplein(2) hasbeenwritten.
Remark1(SeealsoBadea(2014))
1.Theaboveresultsreferredtoproblemsin W 1; withDirichletboundaryconditions,buttheyalsoholdforNeumannormixedboundaryconditions.
3.Theanalysisandtheestimationsoftheglobalconvergenceratewhicharegiven abovereferstotwosidedobstacleproblemswhicharisefromtheminimization offunctionalsdefinedon W 1; , 1< < 1
4.Wecancomparetheconvergencerateswehaveobtainedwithsimilaronesinthe literatureinthecaseof H 1 (p D q D 2)and d D 2.Inthiscase,wegetthat theglobalconvergencerateofAlgorithm 2 is 1 1 1CCJ 3 .Thesameestimate,of 1 1 1CCJ 3 ,isobtainedbyR.Kornhuberfortheasymptoticconvergencerateof thestandardmonotonemultigridmethods forthecomplementarityproblems.
TheresultsinthissectionaredetailedinBadeaandKrause(2012)whereone-and two-levelmethodshavebeenintroducedandanalyzedforthesecondkindandquasivariationalinequalities.Inthecaseofthevariationalinequalitiesofthesecondkind, let ' W K ! R beaconvex,lowersemicontinuous,notdifferentiablefunctionaland, if K isnotbounded,weassumethat F C ' iscoercive,i.e. F .v/ C '.v/ !1,as kv k!1;v 2 K .Weconsiderthevariationalofthesecondkind
u 2 K : hF 0 .u/;v uiC '.v/ '.u/ 0; forany v 2 K (3) which,inviewofthepropertiesof F and ' ,hasauniquesolution.Anexample ofsuchaproblemisgivenbythecontactproblemswithTrescafriction.Tosolve problem(3),weintroduce
Algorithm3 Westartthealgorithmwithanarbitraryu0 2 K.Atiterationn C 1,havingun 2 K,n 0,wecomputesequentiallyfori D 1; ; m,thelocal correctionswnC1 i 2 Vi ; unC i 1
C wnC1 i 2 Kasthesolutionofthevariational inequality
forany vi 2 Vi ; unC i 1 m C vi 2 K,andthenweupdateunC i m D unC
for v; w 2 K ,and vi ; wi 2 Vi , i D 1;:::; m,inAssumption 1.Ingeneral, ' has notsuchapropertyandtoshowthatthisassumptionholdswhenthefiniteelement spacesareused,wehavetotakeanumericalapproximationof ' .Theconvergence ofAlgorithm 3 isprovedbythefollowing
Theorem3 UndertheaboveassumptionsonV,Fand ' ,letubethesolutionof theproblemandun ,n 0,beitsapproximationsobtainedfromAlgorithm 3.If Assumption 1 holds,thenthereexistsM >0 suchthatsuchthat kunC i m k M, n 0;1 i m,andwehavethefollowingerrorestimations:
Inthecaseofthequasivariationalinequalities,weconsideronlythecaseof p D q D 2 andlet ' W K K ! R beafunctionalsuchthat,forany u
, '.u; / W K ! R isconvex,lowersemicontinuousand,if K isnotbounded, F . / C '.u; / is coercive,i.e. F .v/ C '.u;v/ !1 as kv k!1;v 2 K .Weassumethatforany M >0 thereexistsaconstant cM >0 suchthat j'.v1 ; w2 / C '.v2 ;
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