偏微分方程的數(shù)值近似法

定　價：￥81.00

作　者：	（意）Alfio Quarteroni，[意]Alberto Valli著
出版社：	世界圖書出版公司北京公司
叢編項：
標　簽：	微積分

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ISBN：	9787506236171	出版時間：	1998-03-01	包裝：	簡裝本
開本：	20cm	頁數(shù)：	543	字數(shù)：

內(nèi)容簡介

　　out their stability and convergence analysis， derive error bounds， and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis， description of algorithms and discussion of applications is our primary concern. Many kinds of problems are addressed： linear and nonlinear， steady and time-dependent， having either smooth or non-smooth solutions. Besides model equations， we consider a number of （initial-） boundary value problems of interest in several fields of applications. Part I is devoted to the description and analysis of general numerical methods for the discretization of partial differential equations.

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圖書目錄

PartI.BasicConceptsandMethodsforPDEs'Approximation
1.Introduction
1.1TheConceptualPathBehindtheApproximation
1.2PreliminaryNotationandFunctionSpaces
1.3SomeResultsAboutSobolevSpaces
1.4ComparisonResults
2.NumericalSolutionofLinearSystems
2.1DirectMethods
2.1.1BandedSystems
2.1.2ErrorAnalysis
2.2GeneralitiesonIterativeMethods
2.3ClassicalIterativeMethods
2.3.1JacobiMethod
2.3.2Gauss:SeidelMethod
2.3.3RelaxationMethods(S.O.R.andS.S.O.R.)
2.3.4ChebyshevAccelerationMethod
2.3.5TheAlternatingDirectionIterativeMethod
2.4ModernIterativeMethods
2.4.1PreconditionedRichardsonMethod
2.4.2ConjugateGradientMethod
2.5Preconditioning
2.6ConjugateGradientandLanczoslikeMethodsfor
Non-SymmetricProblems
2.6.1GCR,OrthominandOrthodirIterations
2.6.2ArnoldiandGMRESIterations
2.6.3Bi-CG,CGSandBi-CGSTABIterations
2.7TheMulti-GridMethod
2.7.1TheMulti-GridCycles
2.7.2ASimpleExample
2.7.3Convergence
2.8Complements
3.FiniteElementApproximation
3.1Triangulation
3.2Piecewise-PolynomialSubspaces
3.2.1TheScalarCase
3.2.2TheVectorCase
3.3DegreesofFreedomandShapeFunctions
3.3.1TheScalarCase:TriangularFiniteElements
3.3.2TheScalarCase:ParallelepipedalFiniteElements
3.3.3TheVectorCase
3.4TheInterpolationOperator
3.4.1InterpolationError:theScalarCase
3.4.2InterpolationError:theVectorCase
3.5ProjectionOperators
3.6Complements
4.PolynomialApproximation
4.1OrthogonalPolynomials
4.2GaussianQuadratureandInterpolation
4.3ChebyshevExpansion
4.3.1ChebyshevPolynomials
4.3.2ChebyshevInterpolation
4.3.3ChebyshevProjections
4.4LegendreExpansion
4.4.1LegendrePolynomials
4.4.2LegendreInterpolation
4.4.3LegendreProjections
4.5Two-DimensionalExtensions
4.5.1TheChebyshevCase
4.5.2TheLegendreCase
4.6Complements
5.Galerkin,CollocationandOtherMethods
5.1AnAbstractReferenceBoundaryValueProblem
5.1.1SomeResultsofFunctionalAnalysis
5.2GalerkinMethod
5.3Petrov-GalerkinMethod
5.4CollocationMethod
5.5GeneralizedGalerkinMethod
5.6Time-AdvancingMethodsforTime-DependentProblems
5.6.1Semi-DiscreteApproximation
5.6.2Fully-DiscreteApproximation
5.7Fractional-StepandOperator-SplittingMethods
5.8Complements
PartII.ApproximationofBoundaryValueProblems
6.EllipticProblems:ApproximationbyGalerkinand
CollocationMethods
6.1ProblemFormulationandMathematicalProperties
6.1.1VariationalFormofBoundaryValueProblems
6.1.2Existence,UniquenessandA-PrioriEstimates
6.1.3RegularityofSolutions
6.1.4OntheDegeneracyoftheConstantsinStability
andErrorEstimates...
6.2NumericalMethods:ConstructionandAnalysis
6.2.1GalerkinMethod:FiniteElementandSpectral
Approximations
6.2.2SpectralCollocationMethod
6.2.3GeneralizedGalerkinMethod
6.3AlgorithmicAspects
6.3.1AlgebraicFormulation
6.3.2TheFiniteElementCase
6.3.3TheSpectralCollocationCase
6.4DomainDecompositionMethods
6.4.1TheSchwarzMethod
6.4.2Iteration-by-SubdomainMethodsBasedon
TransmissionConditionsattheInterface
6.4.3TheSteklov-PoincareOperator
6.4.4TheConnectionBetweenIterations-by-Subdomain
MethodsandtheSchurComplementSystem
7.EllipticProblems:ApproximationbyMixedand
HybridMethods
7.1AlternativeMathematicalFormulations
7.1.1TheMinimumComplementaryEnergyPrinciple
7.1.2Saddle-PointFormulations:MixedandHybrid
Methods
7.2ApproximationbyMixedMethods
7.2.1SettingupandAnalysis
7.2.2AnExample:theRaviart-ThomasFiniteElements
7.3SomeRemarksontheAlgorithmicAspects
2.4TheApproximationofMoreGeneralConstrained
Problems
7.4.1AbstractFormulation
7.4.2AnalysisofStabilityandConvergence
7.4.3HowtoVerifytheUniformCompatibilityCondition
7.5Complements
8.SteadyAdvection-DiffusionProblems
8.1MathematicalFormulation
8.2AOne-DimensionalExample
8.2.1GalerkinApproximationandCenteredFinite
Differences
8.2.2UpwindFiniteDifferencesandNumericalDiffusion
8.2.3SpectralApproximation
8.3StabilizationMethods
8.3.1TheArtificialDiffusionMethod
8.3.2StronglyConsistentStabilizationMethodsfor
FiniteElements
8.3.3StabilizationbyBubbleFunctions
8.3.4StabilizationMethodsforSpectralApproximation
8.4AnalysisofStronglyConsistentStabilizationMethods
8.5SomeNumericalResults
8.6TheHeterogeneousMethod
9.TheStokesProblem
9.1MathematicalFormulationandAnalysis
9.2GalerkinApproximation
9.2.1AlgebraicFormoftheStokesProblem
9.2.2CompatibilityConditionandSpuriousPressure
Modes
9.2.3Divergence-FreePropertyandLockingPhenomena
9.3FiniteElementApproximation
9.3.1DiscontinuousPressureFiniteElements
9.3.2ContinuousPressureFiniteElements
9:4StabilizationProcedures
9.5ApproximationbySpectralMethods
9.5.1SpectralGalerkinApproximation
9.5.2SpectralCollocationApproximation
9.5.3SpectralGeneralizedGalerkinApproximation
9.6SolvingtheStokesSystem
9.6.1ThePressure-MatrixMethod
9.6.2TheUzawaMethod
9.6.3TheArrow-HurwiczMethod
9.6.4PenaltyMethods
9.6.5TheAugmented-LagrangianMethod
9.6.6MethodsBasedonPressureSoIvers
9.6.7AGlobalPreconditioningTechnique
9.7Complements
10.TheSteadyNavier-StokesProblem
10.1MathematicalFormulation
10.1.1OtherKindofBoundaryConditions
10.1.2AnAbstractFormulation
10.2FiniteDimensionalApproximation
10.2.1AnAbstractApproximateProblem
10.2.2ApproximationbyMixedFiniteElementMethods
10.2.3ApproximationbySpectralCollocationMethods
10.3NumericalAlgorithms
10.3.1NewtonMethodsandtheContinuationMethod
10.3.2AnOperator-SplittingAlgorithm
10.4StreamFunction-VorticityFormulationofthe
Navier-StokesEquations
10.5Complements
PartIII.ApproximationofInitial-BoundaryValueProblems
11.ParabolicProblems
11.1Initial-BoundaryValueProblemsandWeakFormulation
11.1.1MathematicalAnalysisofInitial-BoundaryValue
Problems
11.2Semi-DiscreteApproximation
11.2.1TheFiniteElementCase
11.2.2TheCaseofSpectralMethods
11.3Time-AdvancingbyFiniteDifferences
11.3.1TheFiniteElementCase
11.3.2TheCaseofSpectralMethods
11.4SomeRemarksontheAlgorithmicAspects
11.5Complements
12.UnsteadyAdvection-DiffusionProblems
12.1MathematicalFormulation
12.2Time-AdvancingbyFiniteDifferences
12.2.1ASharpStabilityResultforthe0-scheme
12.2.2ASemi-ImplicitScheme
12.3TheDiscontinuousGalerkinMethodforStabilized
Problems
12.4Operator-SplittingMethods
12.5ACharacteristicGalerkinMethod
13.TheUnsteadyNavier-StokesProblem
13.1TheNavier-StokesEquationsforCompressibleand
IncompressibleFlows
13.1.1CompressibleFlows
13.1.2IncompressibleFlows
13.2MathematicalFormulationandBehaviourofSolutions
13.3Semi-DiscreteApproximation
13.4Time-AdvancingbyFiniteDifferences
13.5Operator-SplittingMethods
13.6OtherApproaches
13.7Complements
14.HyperbolicProblems
14.1SomeInstancesofHyperbolicEquations
14.1.1LinearScalarAdvectionEquations
14.1.2LinearHyperbolicSystems
14.1.3Initial-BoundaryValueProblems
14.1.4NonlinearScalarEquations
14.2ApproximationbyFiniteDifferences
14.2.1LinearScalarAdvectionEquationsandHyperbolic
Systems
14.2.2Stability,Consistency,Convergence
14.2.3NonlinearScalarEquations
14.2.4HighOrderShockCapturingSchemes
14.3ApproximationbyFiniteElements
14.3.1GalerkinMethod
14.3.2StabilizationoftheGalerkinMethod
14.3.3Space-DiscontinuousGalerkinMethod
14.3.4SchemesforTime-Discretization
14.4ApproximationbySpectralMethods
14.4.1SpectralCollocationMethod:theScalarCase
14.4.2SpectralCollocationMethod:theVectorCase
14.4.3Time-AdvancingandSmoothingProcedures
14.5SecondOrderLinearHyperbolicProblems
14.6TheFiniteVolumeMethod
14.7Complements
References
SubjectIndex