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1、上海交通大学硕士学位论文混合型方程的DG有限元方法姓名：张天翼申请学位级别：硕士专业：计算数学指导教师：黄建国20081201上海交通大学硕士学位论文DISCONTINUOUSGALERKINFINITEELEMENTMETHODFORMIXED-TYPEEQUATIONABSTRACTInthetheoryoflinearpartialdi®erentialequations,thedistinctionamonghyperbolic,parabolicandellipticequations
2、issofundamentalthatitrequireseachtypeofequationstomeetspeci¯cboundaryorinitialconditionstoassureexistenceanduniquenessofsolutions,andtoguaranteethewell-posednessoftheproblem.Thisdistinctionmakesitdi±culttostudymixed-typeequations,whicharehyperboliceq
3、uationsinsomesubregionsandellipticequationsinothers.Anditisusuallydi±culttoknowtheproperboundaryconditionstoensuretheexistenceanduniquenessofsolutions.DuetoFriedrichsin'SymmetricPositiveLinearDi®erentialEquation',onecanrewritemixed-typeequationsinsym
4、metricpositiveformsinordertoimposefeasibleboundaryconditions.TheTricomiequationisamixed-typeequation,whichrepresentsasahyperbolicequationintheupperhalfplaneandanellipticequationinthelowerhalfplane.IthasbeenshownthattheTricomiequationcanbeexpressedins
5、ymmetricpositiveform.Admissibleboundaryconditionscanbederivedtoensuretheexistenceanduniquenessofthesolution.BasedonthediscontinuousGalerkinFiniteElementMethod(DGFEM)forhyperbolicequationsandellipticequations,itcanalsobeappliedtomixed-typeequationsins
6、ymmetricpositiveforms.ThisthesisshowswhatboundaryconditionsareadmissiblefortheTricomiequationforanyregionwithpiecewisesmoothboundaries,andconstructstherelevantDGFEMfortheequation.Throughconstructingboundaryoperatorandinterfaceoperator,andusingaverage
7、-ii-上海交通大学硕士学位论文operatorandjumpoperator,onecande¯nethePrimalFormulationofDGFEMbasedonshape-regulartriangulationoftheregionandderivethebilinearformoftheDGFEM.Bygivingproperapproximationfunctionspaceandnormsandseminorms,onecanstudytheL-coercivity,stabi
8、lity,continuity,consistencyandconvergenceofthePrimalFormulationofDGFEM.Italsogivesaprioriestimateforthesolutionwithregularityresults.KEYWORDS:mixed-typeEquations,theTricomiEquation,SymmetricPosi-tiveForm,DGFEM-iii-上海交通大学学位论文原创性声明本人郑重声明：所呈交的学位论文，是本人在导
