Domain decomposition methods : algorithms and theory = 区域分解算法 : 算法与理论 /

introduction.

1.1basicideasofdomaindecomposition

1.2matrixandvectorrepresentations

1.3nonoverlappingmethods

1.3.1anequationforur:theschurcomplementsystem

1.3.2anequationfortheflux

1.3.3thedirichlet-neumannalgorithm

1.3.4theneumann-neumannalgorithm

1.3.5adirichlet-dirichletalgorithmorafetimethod

1.3.6thecaseofmanysubdomains

1.4theschwarzalternatingmethod

1.4.1descriptionofthemethod

1.4.2theschwarzalternatingmethodasarichardsonmethod

1.5blockjacobipreconditioners

1.6someresultsonschwarzalternatingmethods

1.6.1analysisforthecaseoftwosubdomains

1.6.2thecaseofmorethantwosubdomainsabstracttheoryofschwarzmethods

2.1introduction

2.2schwarzmethods

2.3convergencetheory

.2.4historicalremarks

2.5additionalresults

2.5.1coloringtechniques

2.5.2.ahybridmethod

2.5.3comparisonresults

2.6remarksontheimplementation

3two-leveloverlappingmethods

3.1introduction

3.2localsolvers

3.3acoarseproblem

3.4scalingandquotientspacearguments

3.5technicaltools

3.6convergenceresults

3.7remarksontheimplementation

3.8numericalresults

3.9restrictedschwarzalgorithms

3.10alternativecoarseproblems

3.10.1convergenceresults

3.10.2smoothedaggregationtechniques

3.10.3partitionofunitycoarsespaces

4substructuringmethods:introduction

4.1introduction

4.2problemsettingandgeometry

4.3schurcomplementsystems

4.4,discreteharmonicextensions

4.5conditionnumberoftheschurcomplement

4.6technicaltools

4.6.1interpolationintocoarsespaces

4.6.2inequalitiesforedges

4.6.3inequalitiesforfaces

4.6.4inequalitiesforverticesandauxiliaryresults

5primaliterativesubstructuringmethods

5.1introduction.

5.2localdesignandanalysis

5.3localsolvers

5.4coarsespacesandconditionnumberestimates

5.4.1vertexbasedmethods

5.4.2wirebasketbasedalgorithms

5.4.3facebasedalgorithms

6neumann-neumannandfetimethods

6.1introduction

6.2balancingneumann-neumannmethods

6.2.1definitionofthealgorithm

6.2.2matrixformofthealgorithm

6.2.3conditionnumberbounds

6.3one-levelfetimethods

6.3.1areviewoftheone-levelfetimethods

6.3.2thecaseofnonredundantlagrangemultipliers

6.3.3thecaseofredundantlagrangemultipliers

6.4dual-primalfetimethods

6.4.1feti-dpmethodsintwodimensions

6.4.2afamilyoffeti-dpalgorithmsinthreedimensions

6.4.3analysisofthreefeti-dpalgorithms

6.4.4implementationoffeti-dpmethods

6.4.5computationalresults

spectralelementmethods

7.1introduction..

7.2deville-mundpreconditioners

7.3two-leveloverlappingschwarzmethods

7.4iterativesubstructuringmethods

7.4.1technicaltools

7.4.2algorithmsandconditionnumberbounds

7.5remarksonpandhpapproximations

7.5.1moregeneralpapproximations

7.5.2extensionstohpapproximations

linearelasticity

8.1introduction..

8.2atwo-leveloverlappingmethod

8.3iterativesubstructuringmethods

8.4awirebasketbasedmethod

8.4.1anextensionfromtheinterface

8.4.2anextensionfromthewirebasket

8.4.3awirebasketpreconditionerforlinearelasticity

8.5neumann-neumannandfetimethods

8.5.1aneumann-neumannalgorithmforlinearelasticity

8.5.2one-levelfetialgorithmsforlinearelasticity

8.5.3feti-dpalgorithmsforlinearelasticity

9preconditionersforsaddlepointproblems

9.1introduction

9.2blockpreconditioners

9.3flowsinporousmedia

9.3.1iterativesubstructuringmethods

9.3.2hybrid-mixedformulationsandspectral

equivalencieswithcrouzeix-raviartapproximations

9.3.3abalancingneumann-neumannmethod

9.3.4overlappingmethods

9.4thestokesproblemandalmostincompressibleelasticity.

9.4.1blockpreconditioners

9.4.2iterativesubstructuringmethods

9.4.3computationalresults

10problemsinh(dlv;)andh(curl;)

10.1overlappingmethods

10.1.1problemsinh(curl;)

10.1.2problemsinh(div;d)

10.1.3finalremarksonoverlappingmethodsand

numericalresults

10.2iterativesubstructuringmethods

10.2.1technicaltools

10.2.2aface-basedmethod

10.2.3aneumann-neumannmethod

10.2.4remarksontwo-dimensionalproblemsand

numericalresults

10.2.5iterativesubstructuringfornede1ecapproximation

inthreedimensions

11indefiniteandnonsymmetricproblems

11.1introduction

11.2algorithmsonoverlappingsubregions

11.3aniterativesubstructuringmethod

11.4numericalresults

11.4.1anonsymmetricproblem

11.4.2thehelmholtzequation

11.4.3avariable-coefficient,nonsymmetricindefiniteproblem

11.5additionaltopics

11.5.1convection-diffusionproblems

11.5.2thehelmholtzequation

11.5.3optimizedinterfaceconditions

11.5.4nonlinearandeigenvalueproblemsaellipticproblemsandsobolevspaces

a.1sobolevspaces

a.2tracespaces

a.3linearoperators

a.4poincareandfriedrichstypeinequalities

a.5spacesofvector-valuedfunctions

a.5.1thespaceh(div)

a.5.2thespaceh(curl;)intwodimensions

a.5.3thespaceh(curl;)inthreedimensions

a.5.4thekernelandrangeofthecurlanddivergenceoperators

a.6positivedefiniteproblems

a.6.1scalarproblems

a.6.2linearelasticity

a.6.3problemsinh(div;d)andh(curl;g)

a.7non-symmetricandindefiniteproblems

a.7.1generalizationsofthelax-milgramlemma

a.7.2saddle-pointproblems

a.8regularityresultsbgalerkinapproximations

b.1finiteelementapproximations

b.i.1triangulations

b.l.2finiteelementspaces

b.1.3symmetric,positivedefiniteproblems

b.l.4non-symmetricandindefiniteproblems

b.2spectralelementapproximations

b.3divergenceandcurlconformingfiniteelements

b.3.1raviart-thomaselements

b.3.2nedelecelementsintwodimensions

b.3.3nedelecelementsinthreedimensions

b.3.4thekernelandrangeofthecurlanddivergenceoperators

b.4saddle-pointproblems

b.4.1finiteelementapproximationsforthestokesproblem

b.4.2spectralelementapproximationsforthestokesproblem

b.4.3finiteelementapproximationsforflowsinporousmedia

b.5inverseinequalities.

b.6matrixrepresentationandconditionnumbercsolutionofalgebraiclinearsystems

c.1eigenvaluesandconditionnumber

c.2directmethods

c.2.1factorizations

c.2.2fill-in

c.3richardsonmethod

c.4steepestdescent

c.5conjugategradientmethod

c.6methodsfornon-symmetricandindefinitesystems

c.6.1thegeneralizedminimalresidualmethod

c.6.2theconjugateresidualmethod

references

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