Two new least-squares mixed finite element procedures for convection-dominated Sobolev equations

作者:ZHANG Jian-song1 YANG Dan-ping2 ZHU Jiang3 1 Department of Computational and Applied Mathematics;China Universityof Petroleum;Tsingdao 266555;China.2 Department of Mathematics;East China Normal University;Shanghai 200062;China.3 Laborat'orio Nacional de Computa?ca?o Cient'?fica;25651-075 Petro'polis;RJ;Brazil. 刊名:Applied Mathematics:A Journal of Chinese Universities(Series B) 上传者:谢坤

【摘要】Two new least-squares mixed finite element procedures are formulated for solving convection-dominated Sobolev equations.Optimal H(div;Ω) × H1(Ω) norms error estimates are derived under the standard mixed finite spaces.Moreover,these two schemes provide the approximate solutions with first-order and second-order accuracy in time increment,respectively.

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1IntroductionSobolevequationsariseinthe?owof?uidsthroughfissuredrock,thermodynamicsandotherapplications.Theyaresomeimportantpartialdi?erentialequationsinpracticaluse.TherearemanynumericaltreatmentsforSobolevequationscanbefound:EwingandZhustudiedtheGalerkinmethodsforthisproblemsin[4,10];SunandYanganalyzedthefinitedi?erenceandfiniteelementstreamline-di?usionmethodsforSobolevequationswithconvectiontermin[7,8].Moreover,therearealotofworksonleast-squaresmethodsforSobolevequationswithorwithoutconvectionterm,e.g.,see[5,6,9].Thepurposeofthispaperistoformulatetwonewleast-squaresmixedfiniteelementproce-duresforsolvingconvection-dominatedSobolevequations.Weanalyzetheconvergenceoftheproceduresandgivethecorrespondingerrorestimates,whichshowthatthetwoschemesyieldtheapproximatesolutionswithoptimalaccuracyinH(div;?)H1(?)norms.Let?beaconvexpolygonaldomaininR2,withaLipschitzcontinuousboundary.Weconsiderthefollowingconvection-dominatedSobolevequations:402Appl.Math.J.ChineseUniv.Vol.26,No.4?c(x)??ut+d(x)u?a(x)ut+b(x)u=f(x,t),(x,t)?J,u(x,t)=0,(x,t)J,u(x,0)=u0(x),x?(1.1)whereJ=(0,T],d(x)=(d1(x),d2(x))T.Forconvenience,weassumethat:thereexistsomeconstantsa1,a2,b1,b2,c1,c2suchthat00,wehavethefollowingerrorestimatemax0nNun?uhnH1(?)+0mnaxNn?hnH(div;?)Khum+hk1+t.(3.3)Now,weproveTheorem3.1.Proof.Using(2.4)and(2.6),wehaveaN(n?hn,wn?whn),(h,vh)=1c(c(wn?1?whn?1)+(n?1?hn?1)+tR1n),cvh+h+tdvh+a1(n?1?hn?1+a(wn?1?whn?1)+tR2n),h+avh+tbvh.(3.4)Fromtheapproximatepropertyoffiniteelementspacesweknowthatthereexistsave

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