Numerical Methods for Partial Differential Eq

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1、NumericalMethodsforPartialDifferentialEquationsEricdeSturlerDepartmentofComputerScienceUniversityofIllinoisatUrbana-Champaign11/19/20031©2002EricdeSturlerWhyMoreGeneralSpacesWenowprovideamoreformalframeworkfortheapproximationofsolutionsofPDEsandthesolutiontocertainminimizat

2、ionproblems.Firstweneedtoextendthespaceoffunctionsoverwhichwework2(sayCab[,]forone-dimensionalsecondorderequations).21.ManyusefulproblemsdonothaveaCsolution2.Approximationoversubspaceofbasisfunctions(say,piecewise2polynomials)maynoteasilygiveapproximationthatisC.3.Usefultod

3、efinebestapproximatesolutionusingorthogonalprojection,whichinturnrequiresthespacetobecomplete.Firstweneedtointroduceanumberofconcepts.11/19/20032©2002EricdeSturlerTheSpaceL2b2DefinethespaceLab[],:=→{}fab[],

4、
fdx<∞with2∫a12bb2innerproductuv,=∫uvdxandnormuu=(∫dx).aaHere∫
dxde

5、notesLebesgueintegral,whichismoregeneralthantheRiemannintegral,butyieldsthesamevaluewhentheRiemannintegralisdefined.Aconsequenceoftheabovechoicefornormandspaceisthatb()2∫uvd−=x0⇒uv=aevenwhenux()=vx()isnottrueforallxa∈[,b].Suchuandvaremembersofequivalenceclass.11/19/20033©20

6、02EricdeSturlerTheSpacexxif≠12Consider(ab==0,1),uxx:→,andvx:→.0ifx=120ifx≠12Then,uv−=11ifx=.221112However,∫∫00()u−=vdx00dx+∫1dx=0.2So,aselementsofLwehaveuv=evenifuv(11)≠().222Wesayuv=almosteverywhere(a.e.)11/19/20034©2002EricdeSturlerTheSpaceFunctionsareeq

7、ualiftheydiffer(only)onasetofmeasurezero.Anydenumerable(countable)sethasmeasurezero.xx2,∈
Letux()=∈1,xab[]andvx()=.1,xa∈[],b
bThen()2∫uvd−=x0aAgain,aselementsofLwehaveuv=eventhoughuv≠for2everyx∈
.Thefunctionsuandvarealsosaidtobemembersofthesameequivalenceclass.Ing

8、eneral,werepresentanequivalenceclassoffunctionsbyitssmoothestmember.11/19/20035©2002EricdeSturlerTheSpaceVerifythatuv,anduforalluvLab,,∈[]satisfytheproperties2fora(real)innerproductandnorm,andLab[,]isavectorspace.2ForanyHCHx∈>:0()wecandefinetheH-innerproduct1bb22uv,=Huvdxan

9、dH-normuHH=(udx).H∫∫aauv,,=vuu≥0ααuv,,=uvuu=⇔=00uu,0≥ααuu=uu,0==ifandonlyifu0uvuv+