The compact discontinuous Galerkin method for near

JSciComput(2013)56:291–318DOI10.1007/s10915-012-9676-6TheCompactDiscontinuousGalerkinMethodforNearlyIncompressibleLinearElasticityXuehaiHuang·JianguoHuangReceived:12March2012/Revised:5September2012/Accepted:10December2012/Publishedonline:12January2013©SpringerScience+BusinessMediaNewYork2013AbstractAcompactdiscontinuousGalerkinmethod(CDG)isdevisedfornearlyincom-pressiblelinearelasticity,throughreplacingthegloballiftingoperatorfordeterminingthenumericaltraceofstresstensorinalocaldiscontinuousGalerkinmethod(cf.Chenetal.,MathProblEng20,2010)bythelocalliftingoperatorandremovingsomejumpingterms.Itpossessesthecompactstencil,thatmeansthedegreesoffreedominoneelementareonlyconnectedtothoseintheimmediateneighboringelements.Optimalerrorestimatesinbrokenenergynorm,H1-normandL2-normarederivedforthemethod,whichareuniformwithrespecttotheLaméconstantλ.Furthermore,weobtainapost-processedH(div)-conformingdisplacementbyprojectingthedisplacementandcorrespondingtraceoftheCDGmethodintotheRaviart–Thomaselementspace,andobtainoptimalerrorestimatesofthisnumericalsolutioninH(div)-seminormandL2-norm,whichareuniformwithrespecttoλ.Aseriesofnumericalresultsareofferedtoillustratethenumericalperformanceofourmethod.KeywordsNearlyincompressiblelinearelasticity·DiscontinuousGalerkinmethod·CDGmethod·Erroranalysis·Post-processing1IntroductionLet⊂Rd(d=2,3)beaboundedpolygonorpolyhedron.Themathematicalmodelofisotropicnearlyincompressiblelinearelasticitycanbedescribedas(cf.[7,19])X.HuangCollegeofMathematicsandInformationScience,WenzhouUniversity,Wenzhou325035,Chinae-mail:[e-mail](B)DepartmentofMathematics,andMOE-LSC,ShanghaiJiaoTongUniversity,Shanghai200240,Chinae-mail:[e-mail]onalScience,E-InstituteofShanghaiUniversities,ShanghaiNormalUniversity,Shanghai,China123292JSciComput(2013)56:291–318⎧⎨⎩Aσ−ε(u)=0in,−∇·σ=fin,u=0on∂,(1.1)whereσ=(σij)d×disthestressfield,u=(u1,···,ud)tthedisplacementfield,ε(u):=(εij(u))d×dthelinearizedstrainfieldwithεij(u):=(∂ui/∂xj+∂uj/∂xi)/2,f=(f1,···,fd)ttheprescribedbodyforce,and∇theusualgradientoperator.Moreover,Aindicatesthefourth-ordercompliancetensordefinedbyAσ=12μσ−λdλ+2μ(trσ)δ,wheretristhetraceoperationofasecond-ordertensor(amatrix),δ:=(δij)d×dtheusualKroneckerdelta,andthepositiveconstantsλ∈(0,+∞)andμ∈[μ0,μ1]aretheLamécoefficientsoftheelasticmaterialunderconsiderationwith0μ0μ1+∞.WhentheLamécoefficientλtendsto+∞,theconvergencerateoflow-orderconformingfiniteelementmethodswilldeteriorate,leadingtotheso-calledvolumelockingphenom-enon(cf.[3–5]).Therefore,itisimportanttodesignrobustnumericalmethodforsolving(1.1).Here,the“robustness”meansthatthenumericalsolutionmustconvergetotheexactsolutionuniformlywithrespecttoλ,andsuchkindofnumericalmethodsareusuallycalledlocking-freemethods.Inhistory,variousnumericalmethodshavebeenproposedtoresolvethedifficultyofvolumelocking,includingp-versionfiniteelementmethodin[30],noncon-formingfiniteelementmethodin[8,23,25,31,35],mixedfiniteelementmethodin[2,27,28].Forh-versionconformingfiniteelementmethod,BabuškaandSuriprovedin[3]thatitislocking-freewheneverthepolynomialorderpofthelocalshapespaceisnolessthan4andlockedforp4ontriangularmeshes.Moreover,thelockingphenomenonwillalsotakeplaceforanyp≥1onquadrilateralmeshes.Duetotheflexibilityofconstructingfeasiblelocalshapefunctionspacesandtheadvantageofcapturingnon-smoothoroscillatorysolutionseffectively,discontinuousGalerkin(DG)methodshavebeenwidelyusedforsolvingvariouspartialdifferentialequations(cf.[16]fordetails).Inthecontextofnearlyincompressiblelinearelasticity,symmetricinteriorpenaltydiscontinuousGalerkin(SIPDG)methodswereproposedandanalyzedin[21,22].Butsuchmethodsrelyonthechoiceofadimensionlesspenaltyparameterinthestabilizedterm,whichcannotbequantifiedintheoryandcanonlybedeterminedbynumericalexperience.Basedonafirst-ordersystemwiththreemathematicalequations,reformulatedfrom(1.1),alocaldiscontinuousGalerkinmethod(LDG)waspresentedanddiscussedin[17].Furthermore,thepreviouspapersuggestedapostprocessingofprojectingthedisplacementandcorrespondingtraceoftheLDGmethodintotheRaviart–Thomaselementspace,fromwhichonecanderiveaH(div)-conformingdisplacement.Wementioninpassingthattheerroranalysisforalltheformermethodsrequiresthatthesolutionofproblem(1.1)isatleastH2-smooth.Ontheotherhand,alocking-freenonsymmetricinteriorpenaltydiscontinuousGalerkin(NIPDG)methodwasdevelopedin[33]withtheerroranalysisestablishedinthesettingofweightedSobolevspaces.Theadaptivealgorithmrelatedtothismethodwasdesignedin[34],andarobustaposteriorierroranalysiswasgivenin[12,34].Veryrecently,theauthorsandtheircollaboratorsproposedin[14]aLDGmethodforthefirst-ordersystem(1.1),forwhichtherearenoquantifiedparameterstobedetermined.However,theerroranalysisdevelopedtheredidnotconsidertheinfluenceofLamécoefficients.Inthispaper,weintendtodeviseandstudyacompactdiscontinuousGalerkinmethod(CDG)fornearlyincompressiblelinearelasticity.ObservethattheDGmethodshavethefollowingsignificantfeature:Thedegreesoffreedominoneelementareusuallyconnected123JSciComput(20