Numerical solution of elliptic problems by the wav

NumericalSolutionofEllipticProblemsbytheWaveletElementMethodC.Canuto,A.TabaccoDipartimentodiMatematica,PolitecnicodiTorino,C.soDucadegliAbruzzi,24,10129,Torino,ItalyE-mail:[e-mail],[e-mail]eundPraktischeMathematik,RWTHAachen,Templergraben55,52056Aachen,GermanyE-mail:[e-mail]d(WEM)isaconstructionofmultiresolutionsystemsandbiorthogonalwaveletsonfairlygeneraldomains.Domaindecompositionandmappingtoareferencedomainallowtheuseoftensorproductsofscalingfunctionsandwaveletsontheunitinterval.Appropriatematchingconditionsacrosstheinterelementboundariesyieldagloballycontinuousbiorthogonalwaveletbasis.Inthispaper,wedetailthegeneralconstructionfortwo{dimensionaldomainsandshowhowtousetheWEMforthenumericalsolutionofellipticPDE’sinanL{shapeddomain.1IntroductionTheconstructionofmultiresolutionsystemsandwaveletsongeneraldomainsandmanifoldsinIRnisacrucialissueforapplyingwaveletmethodstothenumericalsolutionofoperatorequationssuchaspartialdierentialandintegralequations.Thisproblemhasbeenrecentlyaddressedbymanyauthors[4,5,7,10,15,16,17,18].In[4],theWaveletElementMethod(WEM)wasintroducedborrowingideasfromanalogousconstructionsinspectralmethods.Tensorproductsofscalingfunctionsandwaveletsontheunitintervalaremappedtothesubdo-mainsinwhichtheoriginaldomainissplit.Bymatchingthesefunctionsacrosstheinterelementboundaries,globallycontinuousbiorthogonalwaveletsystemsareobtained,whichallowthecharacterizationofcertainfunctionspacesandtheirduals.Thesespacescontainfunctionswhichhaveaprescribedsmooth-nessineachsubdomain(withrespecttoaSobolevorBesovscale),andsatisfysuitablematchingconditionsattheinterfaces.While[4]dealswiththegeneralspatialdimensionn,herewefocusontwo{dimensionaldomainsandwederivethecorrespondingmatchingcondi-1tions.Moreover,weconsiderarstexampleofapplicationoftheWEMtothenumericalsolutionofellipticPDE’sindomainswhicharenottheimageofasinglesquare.Tothisend,weconsideranL{shapeddomain.Weexplicitlyderivethematchingcoecientsforthewaveletbasis;next,wepresentsomenumericalresultsthatshowthefeasibilityoftheproposedmethodandconrmthetheoreticallyexpectedfeaturesofthemultiscalebasis.Theoutlineofthepaperisasfollows.InSection2wedescribebiorthog-onalsystemsondomainswhicharethesmoothimageofasinglesquare.InSections3and4,wereviewtheconstructionofmatchedscalingfunctionsandwavelets,respectively.Section5isdevotedtothetreatmentofboundaryconditions.Finally,Section6describestheapplicationoftheWEMtotheL{shapeddomain.Throughoutthepaper,wewillfrequentlyusethefollowingnotation:byABwedenotethefactthatAcanbeboundedbyamultipleconstanttimesB,wheretheconstantisindependentofthevariousparametersAandBmaydependon.Furthermore,ABA(withdierentconstants,ofcourse)willbeabbreviatedbyAB.2BiorthogonalsystemsonsimpledomainsInthissection,weusetensorproductsandmappingstoconstructscalingfunctionsandbiorthogonalwaveletsonsmoothimagesofasquare,startingfromsuitablemultiresolutionanalysesontheunitinterval.2.1Theinterval[0;1]DualmultiresolutionanalysesinL2(0;1)canbeconstructedasfollows(see[1,9,13,19]).Westartfromtwofamiliesofscalingfunctionsj:=fj;k:k2jg;~j:=f~j;k:k2jgL2(0;1);wherej(greaterorequaltosomesuitablej0)isthelevelindexandj:=fj;1;:::;j;Kjgwith0=j;1j;2j;Kj=1:(1)So,eachbasisfunctionisassociatedwithanode,orgridpoint,intheinterval[0;1];theactualpositionoftheinternalnodesj;2;:::;j;Kj1willbeirrelevantinthesequel,exceptthatitisrequiredthatjj+1(seei)below).Itwillalsobeconvenienttoconsiderjasthecolumnvector(j;k)k2j,andanalogouslyforothersetsoffunctions.SettingSj:=spanj,~Sj:=span~j;theconditionsa)-l)listedinProperty2.1arefullled:2Property2.1a)Thesystemsjand~jarerenable,i.e.,thereexistmatricesMj;~Mj,suchthatj=Mjj+1,~j=~Mj~j+1.Thisimplies,inparticular,thattheinducedspacesSj,~Sjarenested,i.e.,SjSj+1,~Sj~Sj+1.b)Thefunctionshavelocalsupport,i.e.,diam(suppj;k)2janddiam(supp~j;k)2j.c)Thesystemsarebiorthogonal,i.e.,(j;k;~j;k0)L2(0;1)=k;k0,forallk;k02j.d)Thesystemsj,~jareuniformlystable,i.e.,forallc:=(ck)k2jXk2jckj;kL2(0;1)kck‘2(j)Xk2jck~j;kL2(0;1):e)Thefunctionsareregular,i.e.,j;k2H(0;1),~j;k2H~(0;1),forsome;~1,whereHs(0;1),s0,denotestheusualSobolevspace.f)ThesystemsareexactoforderL,~L1,respectively,i.e.,algebraicpolynomialsuptothedegreeL1,~L1arereproducedexactly:IPL1(0;1)Sj,IP~L1(0;1)~Sj.g)ThesystemjfulllsaJackson{typeinequality:infvj2SjkvvjkL2(0;1)2sjkvkHs(0;1);v2Hs(0;1);0smin(L;);andaBernstein{typeinequality:kvjkHs(0;1)2jskvjkL2(0;1);vj2Sj;0s:Similarinequalitiesholdfor~jwiththeobviouschangesoftheparameters.h)ThereexistbiorthogonalcomplementspacesTjand~TjsuchthatSj+1=SjTj;Tj?~Sj;~Sj+1=~Sj~Tj;~Tj?Sj:i)ThespacesTjand~Tjhavebasesj=fj;h:h2rjg,~j=f~j;h:h2rjg,(withrj:=j+1nj=fj;1;:::;j;Mjg,0j;1j;Mj1)whicharebiorthogonal(inthesenseofc))anduniformlystable(inthesenseofd)).Thesebasisfunctionsarecalledbiorthogonalwavelets.j)Thecollectionsofthesefunctionsforalljj0formRieszbasesofL2(0;1)