Thiseditioncontainsfournewsectionsonthefollowingtopics：theBDDCdomaindecompositionpreconditioner（Section7.8）,aconvergentadaptivealgorithm（Section9.5）,interiorpenaltymethods（Section10.5）andPoincare-FriedrichsinequalitiesforpiecewiseWp1functions（Section10.6）.Wehavemadeimprovementsthroughoutthetext,manyofwhichweresuggestedbycolleagues,towhomwearegrateful.Newexerciseshavebeenaddedandthelistofreferenceshasalsobeenexpandedandupdated. seriesprefaceprefacetothethirdeditionprefacetothesecondeditionprefacetothefirstedition0basicconcepts0.1weakformulationofboundaryvalueproblems0.2ritz-galerkinapproximation0.3errorestimates0.4piecewisepolynomialspaces-thefiniteelementmethod0.5relationshiptodifferencemethods0.6computerimplementationoffiniteelementmethods0.7localestimates0.8adaptiveapproximation0.9weightednormestimates0.xexercises1sobolevspaces1.1reviewoflebesgueintegrationtheory1.2generalized（weak）derivatives1.3sobolevnormsandassociatedspaces1.4inclusionrelationsandsobolevsinequality1.5reviewofchapter01.6tracetheorems1.7negativenormsandduality1.xexercises2variationalformulationofellipticboundaryvalueproblems2.1inner-productspaces2.2hilbertspaces2.3projectionsontosubspaces2.4rieszrepresentationtheorem2.5formulationofsymmetricvariationalproblems2.6formulationofnonsymmetricvariationalproblems2.7thelax-milgramtheorem2.8estimatesforgeneralfiniteelementapproximation2.9higher-dimensionalexamples2.xexercises3theconstructionofafiniteelementspace3.1thefiniteelement3.2triangularfiniteelementsthelagrangeelementthehermiteelementtheargyriselement3.3theinterpolant3.4equivalenceofelements3.5rectangularelementstensorproductelementstheserendipityelement3.6higher-dimensionalelements3.7exoticelements3.xexercises4polynomialapproximationtheoryinsobolevspaces4.1averagedtaylorpolynomials4.2errorrepresentation4.3boundsforrieszpotentials4.4boundsfortheinterpolationerror4.5inverseestimates4.6tensor.productpolynomialapproximation4.7isoparametricpolynomialapproximation4.8interpolationofnon-smoothfunctions4.9adiscretesobolevinequality4.xexercises5n-dimensionalvariationalproblems5.1variationalformulationofpoissonsequation5.2variationalformulationofthepureneumannproblem5.3coercivityofthevariationalproblem5.4variationalapproximationofpoissonsequation5.5ellipticregularityestimates5.6generalsecond-orderellipticoperators5.7variationalapproximationofgeneralellipticproblems5.8negative-normestimates5.9theplate-bendingbiharmonicproblem5.xexercises6finiteelementmultigridmethods6.1amodelproblem6.2mesh-dependentnorms6.3themultigridalgorithm6.4approximationproperty6.5w-cycleconvergenceforthekthleveliteration6.6]/-cycleconvergenceforthekthleveliteration6.7fullmultigridconvergenceanalysisandworkestimates6.xexercises7additiveschwarzpreconditioners7.1abstractadditiveschwarzframework7.2thehierarchicalbasispreconditioner7.3thebpxpreconditioner7.4thetwo-leveladditiveschwarzpreconditioner7.5nonoverlappingdomaindecompositionmethods7.6thebp