[内容简介]
《偏微分方程与数值方法》的作者Stig Larsson现任瑞典Chalmers大学数学系教授、瑞典科学院院士。本书将微分方程的数学分析及有限差分理论和有限元方法结合起来，讲述线性偏微分方程的基本理论及其常用的数值解法。分别用三章阐述椭圆型、抛物型及双曲型偏微分方程，一章关于其数学理论，一章关于其有限差分方法，一章关于其有限元方法。在论述椭圆型方程之前，讲述常微分方程的两点边值问题；类似地，在论述抛物型和双曲型发展问题之前，讲述常微分方程的初值问题。另有一章研究椭圆型特征值问题和特征函数的展开。附录提供了阅读本书所要求的线性泛函分析及索伯列天空间的背景知识。阅读本书不需要高深的数学分析和泛函分析知识。.
本书适用于应用数学专业和工程专业的高年级本科生和低年级研究生。...
[目次]
1Introduction
1.1Background
1.2NotationandMathematicalPreliminaries
1.3PhysicalDerivationoftheHeatEquation
1.4Problems
2ATwo-PointBoundaryValueProblem
2.1TheMaximumPrinciple
2.2Green'sFunction
2.3VariationalFormulation
2.4Problems
3EllipticEquations
3.1Preliminaries
3.2AMaximumPrinciple
3.3Dirichlet'sProblemforaDisc.Poisson'sIntegral
3.4FundamentalSolutions.Green'sFunction
3.5VariationalFormulationoftheDirichletProblem
3.6ANeumannProblem
3.7Regularity
3.8Problems
4FiniteDifferenceMethodsforEllipticEquations
4.1ATwo-PointBoundaryValueProblem
4.2Poisson'sEquation
4.3Problems
5FiniteElementMethodsforEllipticEquations
5.1ATwo-PointboundaryValueProblem
5.2AModelProbleminthePlane
5.3SomeFactsfromApproximationTheory
5.4ErrorEstimates
5.5AnAPosterioriErrorEstimate
5.6NumericalIntegration
5.7AMixedFiniteElementMethod
5.8Problems
6TheEllipticEigenvalueProblem
6.1EigenfunctionExpansions
6.2NumericalSolutionoftheEigenvalueProblem
6.3Problems
7Initial-ValueProblemsforODEs
7.1TheInitialValueProblemforaLinearSystem
7.2NumericalSolutionofODEs
7.3Problems
8ParabolicEquations
8.1ThePureInitialValueProblem
8.2SolutionbyEigenfunctionExpansion
8.3VariationalFormulation.EnergyEstimates
8.4AMaximumPrinciple
8.5Problems
9FiniteDifferenceMethodsforParabolicProblems
9.1ThePureInitialValueProblem
9.2TheMixedInitial-BoundaryValueProblem
9.3Problems
10TheFiniteElementMethodforaParabolicProblem
10.1TheSemidiscreteGalerkinFiniteElementMethod
10.2SomeCompletelyDiscreteSchemes
10.3Problems
11HyperbolicEquations
11.1CharacteristicDirectionsandSurfaces
11.2TheWaveEquation
11.3FirstOrderScalarEquation
11.4SymmetricHyperbolicSystems
11.5Problems
12FiniteDifferenceMethodsforHyperbolicEquations
12.1FirstOrderScalarEquations
12.2SymmetricHyperbolicSystems
12.3TheWendroffBoxScheme
12.4Problems
13TheFiniteElementMethodforHyperbolicEquations
13.1TheWaveEquation
13.2FirstOrderHyperbolicEquations
13.3Problems
14SomeOtherClassesofNumericalMethods
14.1CollocationMethods
14.2SpectralMethods
14.3FiniteVolumeMethods
14.4BoundaryElementMethods
14.5Problems
ASomeToolsfromMathematicalAnalysis
A.1AbstractLinearSpaces
A.2FunctionSpaces
A.3TheFourierTransform
A.4Problems
BOrientationonNumericalLinearAlgebra
B.1DirectMethods
B.2IterativeMethods.Relaxation,Overrelaxation,andAcceleration
B.3PreconditionedConjugateGradientMethods
B.4PreconditionedConjugateGradientMethods
B.5MultigridandDomainDecompositionMethods
Bibliography
Indexdex