有限元超收敛分析及后验误差估计

Chapter1Basicframework

1.1Preliminaries

1.2Modelproblem

1.3Integralidentity

1.4Globalsuperconvergenceanalysis

1.4.1Supercloseanalysis

1.4.2Globalsuperconvergence

1.4.3Aposteriorierrorestimate

1.5Briefsummaryandnotes

Chapter2Integralidentities

2.1Bilinearrectangularelement

2.1.1Integralidentityfor∫Ω(u-uI)xvx

2.1.2Integralidentityfor∫Ω(u-uI)xvy

2.1.3Integralidentitiesfor∫Ω(u-uI)xvand∫Ω(u-uI)v

2.1.4Summary

2.2Generalresultsforbilinearelements

2.2.1Generalellipticbilinearformofordertwo

2.2.2Bilinearfiniteelementongeneraldomain

2.2.3Regularlocallyrefinedmesh

2.3RectangularLagrangeelementsoforder

2.3.1Integralidentityfor∫Ω(u-uI)xvx(p=2)

2.3.2Integralidentityfor∫Ω(u-uI)xvx(p≥3)

2.3.3Integralidentityfor∫Ω(u-uI)xvy

2.3.4Integralidentityfor∫Ω(u-uI)xvand∫Ωγ(u-uI)v

2.3.5Generalellipticbilinearform

2.4Rectangularfiniteelementswithderivativedegreesoffreedom

2.4.1BicubicHermitedement

2.4.2Adinielement

2.5Rectangularmixedfiniteelements

2.5.1MixedfiniteelementforStokesequation

2.5.2Mixedfiniteelementforellipticequation

2.5.3MixedfiniteelementforMaxwellequation

2.6Summaryofintegralidentities

Chapter3SuperconvergenceAnalysis

3.1Ellipticpartialdifferentialequations

3.1.1Poisson’sequation

3.1.2Ellipticequationofordertwo(Lagrangeelement)

3.1.3

3.4.1Standardfiniteelementscheme

3.4.2Streamlinediffusionmethod

3.4.3DiscontinuousGalerkinmethod

3.5Mixedfinitedements

3.5.1EllipticPDEofordertwo

3.5.2EllipticPDEoforderfour

3.5.3Stokesequation

3.5.4Maxwdl’sequation

3.6Integralequations

3.7Optimalcontrolproblems

3.8Summaryofsuperconvergenceanalysis

Chapter4Morediscussionsonhighaccuracyanalysis

4.1Globalsuperconvergence

4.1.1Interpolationpost-processing

4.1.2Globalsuperconvergence

4.1.3Summaryofglobalsuperconvergence

4.2Extrapolation

4.3Defectcorrection

4.4Localsuperconvergence

4.5Ultraconvergence

4.6Eigenvalueproblems

4.7Numericalexamples

Chapter5Aposteriorierrorestimates

5.1Introduction

5.2Residualtypeaposteriorierrorestimate

5.3Recoverytypeaposteriorierrorestimate

5.4Equivalenceofrecoverytypeestimator

5.5Asymptoticallyexactnessofrecoverytypeestimator

5.6Someremarksontwokindsofestimators

5.7Aposteriorierrorestimateforoptimalcontrolproblems

5.7.1Modelproblem

5.7.2Residualtypeaposteriorierrorestimate

5.7.3Recoverytypeaposteriorierrorestimate

5.8Numericalexamples

Bibliography

　　Thisbooktriestosummarizemostoftheresearchresultsonglobalsuperconvergenceanalysiscompletedbytheauthorandhercolleagues,especiallybyProfessorQ.Lin’sgroup.InsteadofusingtheGreenfunctiontheoryasthetheoreticalbasisasinmostotherearlierbooksonsuperconvergence,ourglobalsuperconvergenceanalysisisbasedontheso-calledintegralidentitytechnique.Foraposteriorierrorestimates,thisbookfocusesontherecoverytypeaposteriorierrorestimate,whichismoreclosetothesuperconvergenceanalysisinitstheoreticalanalysis.Theemphasesandselectionofthetopicsreflectsour(authorandcolleagues)involvementinthefieldoverthepast15years.