有限元方法的数学理论

SeriesPreface

PrefacetotheSecondEdition

PrefacetotheFirstEdition

0BasicConcepts

0.1WeakFormulationofBoundaryValueProblems

0.2Ritz-GalerkinApproximation

0.3ErrorEstimates

0.4PiecewisePolynomialSpaces-TheFiniteElementMethod

0.5RelationshiptoDifferenceMethods

0.6ComputerImplementationofFiniteElementMethods

0.7LocalEstimates

0.8AdaptiveApproximation

0.9WeightedNormEstimates

0.xExercises

1SobolevSpaces

1.1ReviewofLebesgueIntegrationTheory

1.2Generalized(Weak)Derivatives

1.3SobolevNormsandAssociatedSpaces

1.4InclusionRelationsandSobolevsInequality

1.5ReviewofChapter0

1.6TraceTheorems

1.7NegativeNormsandDuality

1.xExercises

2VariationalFormulationofEllipticBoundaryValueProblems

2.1Inner-ProductSpaces

2.2HilbertSpaces

2.3ProjectionsontoSubspaces

2.4RieszRepresentationTheorem

2.5FormulationofSymmetricVariationalProblems

2.6FormulationofNonsymmetricVariationalProblems

2.7TheLax-MilgramTheorem

2.8EstimatesforGeneralFiniteElementApproximation

2.9Higher-dimensionalExamples

2.xExercises

3TheConstructionofaFiniteElementSpace

3.1TheFiniteElement

3.2TriangularFiniteElements

TheLagrangeElement

TheHermiteElement

TheArgyrisElement

3.3TheInterpolant

3.4EquivalenceofElements

3.5RectangularElements

TensorProductElements

TheSerendipityElement

3.6Higher-dimensionalElements

3.7ExoticElements

3.xExercises

4PolynomialApproximationTheoryinSobolevSpaces

4.1AveragedTaylorPolynomials

4.2ErrorRepresentation

4.3BoundsforRieszPotentials

4.4BoundsfortheInterpolationError

4.5InverseEstimates

4.6Tensor-productPolynomialApproximation

4.7IsoparametricPolynomialApproximation

4.8InterpolationofNon-smoothFunctions

4.9ADiscreteSobolevInequality

4.xExercises

5n-DimensionalVariationalProblems

5.1VariationalFormulationofPoissonsEquation.

5.2VariationalFormulationofthePureNeumannProblem.

5.3CoercivityoftheVariationalProblem

5.4VariationalApproximationofPoissonsEquation

5.5EllipticRegularityEstimates

5.6GeneralSecond-OrderEllipticOperators

5.7VariationalApproximationofGeneralEllipticProblems.

5.8Negative-NormEstimates

5.9ThePlate-BendingBiharmonicProblem

5.xExercises

6FiniteElementMultigridMethods

6.1AModelProblem

6.2Mesh-DependentNorms

6.3TheMultigridAlgorithm

6.4ApproximationProperty

6.5W-cycleConvergenceforthekthLevelIteration

6.6V-cycleConvergenceforthekthLevelIteration

6.7FullMultigridConvergenceAnalysisandWorkEstimates

6.xExercises

7AdditiveSchwarzPreconditioners

7.1AbstractAdditiveSchwarzFramework

7.2TheHierarchicalBasisPreconditioner

7.3TheBPXPreconditioner

7.4TheTwo-levelAdditiveSchwarzPreconditioner

7.5NonoverlappingDomainDecompositionMethods

7.6TheBPSPreconditioner

7.7TheNeumann-NeumannPreconditioner

7.xExercises

8Max-normEstimates

8.1MainTheorem

8.2ReductiontoWeightedEstimates

8.3ProofofLemma8.2.6

8.4ProofsofLemmas8.3.7and8.3.11

8.5LpEstimates(RegularCoefficients)

8.6LpEstimates(IrregularCoefficients)

8.7ANonlinearExample

8.xExercises

9AdaptiveMeshes

9.1AprioriEstimates

9.2ErrorEstimators

9.3LocalErrorEstimates

9.4EstimatorsforLinearFormsandOtherNorms

9.5ConditioningofFiniteElementEquations

9.6BoundsontheConditionNumber

9.7ApplicationstotheConjugate-GradientMethod

9.xExercises

10VariationalCrimes

10.1DeparturefromtheFramework

10.2FiniteElementswithInterpolatedBoundaryConditions.

10.3NonconformingFiniteElements

10.4IsoparametricFiniteElements

10.xExercises

11ApplicationstoPlanarElasticity

11.1TheBoundaryValueProblems

11.2WeakFormulationandKornsInequality

11.3FiniteElementApproximationandLocking

11.4ARobustMethodforthePureDisplacementProblem..

11.xExercises

12MixedMethods

12.1ExamplesofMixedVariationalFormulations

12.2AbstractMixedFormulation

12.3DiscreteMixedFormulation

12.4ConvergenceResultsforVelocityApproximation

12.5TheDiscreteInf-SupCondition

12.6VerificationoftheInf-SupCondition

12.xExercises

13IterativeTechniquesforMixedMethods

13.1IteratedPenaltyMethod

13.2StoppingCriteria

13.3AugmentedLagrangianMethod

13.4ApplicationtotheNavier-StokesEquations

13.5ComputationalExamples

13.xExercises

14ApplicationsofOperator-InterpolationTheory

14.1TheRealMethodofInterpolation

14.2RealInterpolationofSobolevSpaces

14.3FiniteElementConvergenceEstimates

14.4TheSimultaneousApproximationTheorem

14.5PreciseCharacterizationsofRegularity

14.xExercises

References

Index

　　有限元法被广泛用于工程设计和工程分析。本书是Springer出版的《应用数学教材》丛书之15。全书分成15章，在第1版的基础上增加了加性Schwarz预条件和自适应格；书中不但提供有限元法系统的数学理论。还兼重在工程设计和分析中的应用算法效率、程序开发和较难的收敛问题。