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1、ANALYSISOFEXPANDEDMIXEDMETHODSFORFOURTH-ORDERELLIPTICPROBLEMS,IZhangxinChenAbstract.Therecentlyproposedexpandedmixedformulationfornumericalso-lutionofsecondorderellipticproblemsishereextendedtofourthorderellipticprob-lems.Thisexpandedformulationforthedi erentialproblemsunde
2、rconsiderationdi ersfromtheclassicalformulationinthatthreevariablesaretreated,i.e.,thedisplacementandthestressandmomenttensors.Itworksforthecasewherethecoecientofthedi erentialequationsissmallanddoesnotneedtobeinverted,orforthecaseinwhichthestresstensoroftheequationsdoesno
3、tneedtobesymmet-ric.Basedonthisnewformulation,variousmixed niteelementsforfourthorderproblemsareconsidered;errorestimatesofquasi-optimaloroptimalorderdependinguponthemixedelementsarederived.Implementationtechniquesforsolvingthelin-earsystemarisingfromtheseexpandedmixedmetho
4、dsarediscussed,andnumericalresultsarepresented.I.INTRODUCTIONRecentlyanewmixedformulationfornumericalsolutionofsecondorderellipticproblemshasbeendevelopedandanalyzed[7,8,9].Thisnewformulationexpandstheclassicalmixedoneinthesensethatthreevariablesareexplicitlytreated,i.e.,th
5、escalarunknown,itsgradient,andits
ux(thecoecienttimesthegradient);thatiswhyitiscalledanexpandedmixedformulation.Itworksforthecasewherethecoecientofthedi erentialequationsissmallanddoesnotneedtobeinverted,soitappliesto
owequationswithlowpermeabilityandtotransportequationsw
6、ithsmalldispersioninreservoirsimulation[11].Theexpandedmixedformulationwas rstanalyzedforlinearsecondorderprob-lems[8],andthenextendedtoquasilinearproblems[7,9].Inthispaperweconsiderthisformulationforfourthorderellipticproblemswithconstantandvariableco-ecients.Forthefourth
7、orderproblems,thepresentformulationdi ersfromtheclassicaloneinthatithandlesthreeunknowns,i.e.,thedisplacementandthestressandmomenttensors.Also,ingeneral,mostoftheclassicalmixed niteelementmethodstreatthesedi erentialproblemssolelywithconstantcoecients[4,13,14,16,18].Usingt
8、hepresentapproach,theproblemassociatedwithvariablecoe-cientscanbeeasilyhandledthr
