李会元——summerschool2

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1、GalerkinSpectralMethodsforEigenvalueProblemsHuiyuanLiInstituteofSoftwareChineseAcademyofSciencesEmail:huiyuan@iscas.ac.cnJuly26,2013H.Li(ISCAS)SpectralMethodJuly26,20131/42SomeimportantissuesLaplacianeigenvaluesusingChebyshevpolynomials6104102N≈26082π10010−210relativeerror=0.01%−410−61

2、0−810−1010−1210−1410−16101000200030004000H.Li(ISCAS)SpectralMethodJuly26,20132/42Someimportantissuesmulti-domainspectralmethodsforproblemswithjumpcoecients1Someimportantissuesmulti-domainspectralmethodsforproblemswithjumpcoecientsmulti-domainspectralmethodsforL-shapedomainproblemSpec

3、tralmethodsforanisotropicproblems2Spectralmethodsontriangles3Spectralmethodsondisks4DiagonalizationandeigenfunctionsH.Li(ISCAS)SpectralMethodJuly26,20133/42Someimportantissuesmulti-domainspectralmethodsforproblemswithjumpcoecientsmulti-domainspectralmethods+problem¢¨¨−(au′)′=inI=(−1;

4、1);¢¨¨1;x∈I1∶=(−1;0);¦a(x)=¦¨¨u(±)=0;¨¨4;x∈I2∶=(0;1):¤¤rukSI∈H(Ii)foranyr.i+domaindecompositionandmulti-domainGalerkinmethodI=I1∪I2;N=2n+n:H.Li(ISCAS)SpectralMethodJuly26,20134/42Someimportantissuesmulti-domainspectralmethodsforproblemswithjumpcoecients010−210−410−610−810−1010−1210−14

5、10−1610012345101010101010exponentialconvergencev.s.2-ndorderconvergence.H.Li(ISCAS)SpectralMethodJuly26,20135/42Someimportantissuesmulti-domainspectralmethodsforL-shapedomainproblem1Someimportantissuesmulti-domainspectralmethodsforproblemswithjumpcoecientsmulti-domainspectralmethodsfo

6、rL-shapedomainproblemSpectralmethodsforanisotropicproblems2Spectralmethodsontriangles3Spectralmethodsondisks4DiagonalizationandeigenfunctionsH.Li(ISCAS)SpectralMethodJuly26,20136/42Someimportantissuesmulti-domainspectralmethodsforL-shapedomainproblemL-shapedomainLaplacianeigenvalue¢¨¨−

7、
u=;in=1∪2∪3;¦¨¨@u=0;on@:¤21=(−1;0)×(0;1);2=(−1;0);3=(0;1)×(−1;0):H.Li(ISCAS)SpectralMethodJuly26,20137/42Someimportantissuesmulti-domainspectralmethodsforL-shapedomainproblemL-shapedomain○2~3270concaveangle,strongsingularitynear(0:0)asr.5~31u1possessthestrongestsingularity:u1∈B()