1 Analyzing Stationary and Periodic Solutions of S

1AnalyzingStationaryandPeriodicSolutionsofSystemsofParabolicPartialDierentialEquationsbyUsingSingularSubspacesasReducedBasisHelmutJarausch1AbstractComputingandanalyzingthebifurcationbehaviourofstationaryorperiodicsolutionsofnonlinearparabolicPDEsleadtoverylarge,parameterdepen-dendnonlinearalgebraicsystemsafterdiscretization.Wedecomposethefull(discrete)spaceintoatinysubspaceanditsorthogonalcomplement.Afterasuitabletransformationthefullsystemrestrictedtothissubspacegivesatinysystemwhichmodelsthenonlinearbehaviourofthefullsystem,thisissimilartothereducedbasistechnique.Ontheotherhandwedonotthrowawaythe(large)complementarysystem,thusthereisnolossofinformation.Thelargesystemislocallycontractive.Thissplittingisobtainedbyausingcertainsingularvectorsofalinearizationofthesystemtospanthetinysub-space.Weavoidusinginvariantsubspaceswhichmaybediulttohandleinthenon{selfadjointcase.Thiscanonlybedoneafterthesystemhasbeentransformedtosomethinglikeanidealnormalequationwhichavoidsthebadconditioningofthestandardnormalequationapproach.Keywords:nonlinearpartialdierentialequations,periodicsolutions,bifurcationanalysis,reducedbasis,Ljapunov{Schmidtre-ductionIntroductionWewanttostudyperiodicandstationarysolutionsofaparameterde-pendentparabolicsystemoftheform(0)_~u=L(~u;)1Inst.f.Geometrieu.Prakt.Mathematik,RWTH,D{52056Aachen,Germany2togetherwithinitialandboundaryconditions.HereL(:;)isanonlinearellipticoperator(system).Discretizing(0)bythemethodoflines|discretizingLbynitedierencesorniteelements|leadstoalargesystemofordinarydif-ferentialequations(10)_u=F(u;)u2Rn2Rpwherethevectorucontainsallthediscretizedcomponentsof~u.Inthispaperwestudy(parameterdependent)stationaryandperi-odicsolutionsof(1’).Thequestionwhether(andtowhatextent)thesesolutions(criticalpoints)approximatethoseofthecontinuousproblem(0)isoutofthescopeofthispaper,seee.g.Brezzietal(1981),andHansbo(1990).Thefactthatequation(1’)arisesfromdiscretizingtheparabolicsys-tem(0)impliesononehandthatthesystem(1’)isverylarge.Ontheotherhandseveralpropertiesofthecontinuoussystemareinheritedbythediscretesystem(1’)|mostimportanttheeigenvaluesofsmallestmodulusandthecorrespondingeigenfunctionsofthecontinuoussystemarewellapproximatedbythediscretesystemforreasonablediscretiza-tions.Themainpracticalprobleminstudyingtheparameterdependenceofsolutionsofthesystem(1’)liesinthehighdimensionalityofthissystem.Aswillbeshowninsection1,bothproblemscanbereducedtoa(large)xedpointequationu=E(u)ThecasewheretheJacobianofE0ofEisselfadjointhasbeenstud-iedbefore,seeJarausch/Mackens(1982,1984,1987)andHetzer/Jarausch/Mackens(1989).InpracticalproblemsE0hasonlyveryfeweigenvalueslargerthan1inmodulus|atleastaftersomepreconditioning.LetPbeanorthogonalprojectorontothedominatingeigenspaceofE0andQitscomplementaryprojector,wegetthesplitsystemp=PE(p+q)q=QE(p+q)withp=Puandq=Qu.3Thep{systemhasonlyveryfewunknownssuchthatexpensivetechniquescanbeappliedforstudyingitssolutionsset.Ontheotherhandtheq{systemcanbemadelocallycontractiveandcanbethussolvedbyxedpointiterations.Theoreticallysolvingtheq{systemandinsertingitssolutionintothep{systemleadstoaLjapunov{Schmidtreductionofthefullsystem.Jarausch/Mackens(1987)solvetheq{systemapproximatelytreatingtheseiterationsas\inneriterationswhereastheiteratesforthep{systemmakeupthe\outeriterations.BycarefullyadaptingtheinneriterationstotheouteriterationsandadjustingtheprecisionoftheprojectorsPandQtheycanprovesuchaschemeconvergent.Thesametechniquecanbeappliedtothemoregeneralsplittingschemepresentedinthispaper.Neglectingtheq{dependenceofthep{systemleadstoalowdimen-sionalapproximatesystemwhichcanberegardedasareducedbasisapproach.Suchanapproachiswellsuitedforstudyingtheparameterdependenceofthesolutionset.Seee.g.Fink/Rheinboldt(1983,1987),Mackens(1988),Porsching(1985),Porsching/LinLee(1987)andRhein-boldt(1986,1988).Anextensiontothecaseofanon{selfadjointJacobianofEhasbeengivenbyShro/Keller(1993).Theysuggestatechniquewhichassumesonlyminimalknowlegdeabouttheunderlyingxedpointequation;theyevendonotneedtoevaluatetheJacobiannorevenitstranspose.Ontheotherhandtheirapproachseemstobelimitedtothecasewherethe\unstablespaceisspannedbyeigenvectors.Furthermoretheirp{andq{systemisnotcompletelydecoupled;thisrestrictstheanalysisofbifurcationphenomenaandtheuseofthep{systemasareducedbasisapproach.Anotherextensiontothenon{selfadjointcasehasbeengiveninJarausch(1991),wherethesplittingisbasedoninvariantsubspacesoftheJacobian.ThereithasbeenobservedthateventheuseofinvariantsubspacesmayleadtodicultiesasitimpliesthattheprojectorsPandQareobliqueingeneral.Furthermoretheapproximationofinvariantsubspacesmaybequitecumbersome.Thepresentapproachmightbelookedatasan\idealnormalequa-tionapproach.Itusessingularsubspacesforsplittingthexedpointequation.Thesearemucheasiertoapproximate,leadtoarbitrarilygooddecouplingofthepartialsystemsandimplyorthogonalprojectorsonly.4Thisapproachreducesquitenaturallytothesplittingwitheigenspacesintheselfadjointcase.Inordertocompute(critical)solutionswithreasonableprecisionitisnotenoughjusttostudy