作者	A.Quarteroni/等
出版社
出版时间	1998-03-01

特色：

片断：1.IntroductionNumericalapproximationofpartialdifferentialequationsisanimportantbranchofNumericalAnalysis.Often,itdemandsaknowledgeofmanyaspectsoftheproblem.Firstofall,thephysicalbackgroundoftheproblemisrequiredinordertounderstandthebehaviourofexpectedsolutions.Thismayoftenleadtothechoiceofconvenientnumericalmethods.Secondly,modernformulationoftheproblembasedonthevariational(weak)formoughttobeconsidered,asitallowsthesearchforgeneralizedsolutionsinHilbert(orBanach)functionalspaces.Variationaltechniquesyielda-prioriestimatesforthesolution,whichinturnindicateinwhichkindofnormsanyvirtualnumericalsolutioncanbeproventobestable.Further-more,resultsaboutsmoothnessofthemathematicalsolutionsmaysuggestthenumericalmethodologytobeused,andconsequently,determinethekindofaccuracythatcanbeachieved.Thelatterispointedoutfromtheerroranalysis.Clearly,specificattentionshouldbepaidtothealgorithmicaspectscon-cernedwiththechoiceofanynumericalmethod.Thisbookaimsatprovidinggeneralideasonnumericalapproximationofpartialdifferentialequations,although(obviously)notallpossibleexistingmethodswillbeconsidered.Inthisrespect,wemainlyfocusonvariationalnumericalmethodsforthediscretizationofspacederivatives,andonfinitedifferenceandfractional-stepmethodsforadvancing,intime,unsteadyprob-lems.Wheneverpossible,wepresenttheunifyingapproachbehinda-prioridif-ferentnumericalstrategies,providegeneraltheoryforanalysisandillustrateavarietyofalgorithmsthatcanbeusedtocomputetheeffectivenumeri-calsolutionoftheproblemathand,takingintoconsiderationitsalgebraicstructure.Consequently,wetrytoavoidusingtechnicalities(ortricks,oralgorithms)thatworkonlyinveryspecificsituations,orthatarenotsus-tainedfromasoundtheoreticalbackground.Someproblems(andmethods)arediscussedonacase-to-casebasis,butveryoftentheyareincludedinasinglelogicalunit(sayChapter,orSection).1.1TheConceptualPathBehindtheApproximationWeconsideragreatnumberofmathematicalproblems,andnumericalmeth-odsfortheirsolution.Fortheapproximationofanygivenboundaryvalueproblem,weschematicallyillustrateinFig.1.1.1thedecisionpaththatneedstobefollowed.Level[1]istheboundaryvalueproblemathandunderitsweakformula-tionaccountingfortheprescribedboundaryconditions.Level[2]providesthekindofdiscretization(ornumericalmethod)thatcanbepursuedinordertoreducethegivenproblemtoonehavingfinitedimension.Ofcourse,thestrategyadoptedwilldeterminethestructureofthenumericalproblem.Throughoutthisbookwemainlyconsidertwokindsofdiscretization.TheformeristheGalerkinmethod,togetherwithitsremarkablevariant,thePetrov-Galerkinmethod,whichisbasedonanintegralformulationofthedifferentialproblem.Theseconddiscretizationweconsider,isthecollocationmethod,whichis,instead,basedonthefulfillmentofthedifferentialequationsatsomeselectedpointsofthecomputationaldomain.WethenreformulatethecollocationmethodunderageneralizedGalerkinmode,preciselycombiningtheGalerkinapproachwithnumericalevaluationofintegralsusingGaussianformulae.Atalowerextent,wewilladdressfinitedifferenceschemesforspacedis-cretization,especiallyfornonlinearconvection-diffusionequationsandforproblemsofwavepropagation.Forthelatterwewillalsopresenttheapproachbasedonthefinitevolumemethod,whichisverypopularincomputationalfluiddynamics.Finally,wewillillustrateshortlytheelementarypnnciplesofthedomaindecompositionmethod,anapproachwhichoffersthebestpromisefortheparallelsolutionoflargeproblemsinthefieldofscientificcomputing.Otherapproachesareoftenencounteredintheliteratureaswell,buttheywillonlybeaddressedincidentallyinthisbook.Level[3]specifiesthenatureofthesubspacesusedintheapproximation.Typically,wehavepiecewise-polynomialfunctionsoflowdegreewhenusingfiniteelements,andglobalalgebraicpolynomialsofhighdegreeforspec-tralmethods.Thesetworemarkablecaseswillbediscussedandanalyzedinsomeoftheirvariants(mixedfiniteelements,LegendreandChebyshevspec-tralcollocationmethods).Thechoiceoperatedatthisleveldeterminesthefunctionalstructureofthenumericalsolution,thekindofaccuracythatcanbeachieved,besidesaffectingthetopologicalformoftheresultingalgebraicsystem.