Generalized difference methods for differential equations : numerical analysis of finite volume methods

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Generalized difference methods for differential equations : numerical analysis of finite volume methods / Ronghua Li, Zhongying Chen, Wei Wu

資料種別:

図書

出版情報:

New York : Marcel Dekker, c2000

形態:

xv, 442 p. ; 24 cm

シリーズ名:

Monographs and textbooks in pure and applied mathematics ; 226 <BA00046446>

著者名:

ISBN:

9780824703301 [0824703308]

書誌ID:

BA45008042

子書誌情報

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書誌詳細

注記:

Bibliography: p. 421-437
Includes index

主題:

Differential equations, Partial--Numerical solutions; Finite diferences

言語:

英語

目次情報:

Preface

Preliminaries / 1：

Sobolev Spaces / 1.1：

Smooth approximations. Fundamental lemma of variational methods / 1.1.1：

Generalized derivatives and Sobolev spaces / 1.1.2：

Imbedding and trace theorems / 1.1.3：

Finite element spaces / 1.1.4：

Interpolation error estimates in Sobolev spaces / 1.1.5：

Variational Problems and Their Approximations / 1.2：

Abstract variational form / 1.2.1：

Green's formulas and variational problems / 1.2.2：

Well-posedness of variational problems / 1.2.3：

Approximation methods. A necessary and sufficient condition for approximate-solvability / 1.2.4：

Galerkin methods / 1.2.5：

Generalized Galerkin methods / 1.2.6：

Bibliography and Comments

Two Point Boundary Value Problems / 2：

Basic Ideas of the Generalized Difference Method / 2.1：

A variational form / 2.1.1：

Generalized Galerkin variational principles / 2.1.2：

Generalized difference methods / 2.1.4：

Linear Element Difference Schemes / 2.2：

Trial and test function spaces / 2.2.1：

Difference equations / 2.2.2：

Convergence estimates / 2.2.3：

Quadratic Element Difference Schemes / 2.3：

Trial and test spaces / 2.3.1：

Convergence order estimates / 2.3.2：

Cubic Element Difference Schemes / 2.4：

Some lemmas / 2.4.1：

Existence, uniqueness and stability / 2.4.4：

Numerical examples / 2.4.5：

Estimates in L[superscript 2] and Maximum Norms / 2.5：

L[superscript 2]-estimates / 2.5.1：

Maximum norm estimates / 2.5.2：

Superconvergence / 2.6：

Optimal stress points / 2.6.1：

Superconvergence for linear element difference schemes / 2.6.2：

Superconvergence for cubic element difference schemes / 2.6.3：

Generalized Difference Methods for a Fourth Order Equation / 2.7：

Generalized difference equations / 2.7.1：

Positive definiteness of a(u[subscript h], II*[subscript h] u[subscript h]) / 2.7.2：

Second Order Elliptic Equations / 2.7.3：

Introduction / 3.1：

Generalized Difference Methods on Triangular Meshes / 3.2：

Generalized difference equation / 3.2.1：

a priori estimates / 3.2.3：

Error estimates / 3.2.4：

Generalized Difference Methods on Quadrilateral Meshes / 3.3：

Numerical example / 3.3.1：

L[superscript 2] and Maximum Norm Estimates / 3.5：

L[superscript 2] estimates / 3.6.1：

A maximum estimate and some remarks / 3.6.2：

Superconvergences / 3.7：

Weak estimate of interpolations / 3.7.1：

Superconvergence estimates / 3.7.2：

Fourth Order and Nonlinear Elliptic Equations / 4：

Mixed Generalized Difference Methods Based on Ciarlet-Raviart Variational Principle / 4.1：

Mixed generalized difference equations / 4.1.1：

Mixed Generalized Difference Methods Based on Hermann-Miyoshi Variational Principle / 4.1.2：

Numerical experiments / 4.2.1：

Nonconforming Generalized Difference Method Based on Zienkiewicz Elements / 4.3：

Variational principle / 4.3.1：

Generalized difference schemes based on Zienkiewicz elements / 4.3.2：

Error analyses / 4.3.3：

Numerical experiment / 4.3.4：

Nonconforming Generalized Difference Methods Based on Adini Elements / 4.4：

Generalized difference scheme / 4.4.1：

Error estimate / 4.4.2：

Second Order Nonlinear Elliptic Equations / 4.4.3：

Parabolic Equations / 4.5.1：

Semi-discrete Generalized Difference Schemes / 5.1：

Problem and schemes / 5.1.1：

L[superscript 2]-error estimate / 5.1.2：

H[superscript 1]-error estimate / 5.1.4：

Fully-discrete Generalized Difference Schemes / 5.2：

Fully-discrete schemes / 5.2.1：

Error estimates for backward Euler generalized difference schemes / 5.2.2：

Error estimates for Crank-Nicolson generalized difference schemes / 5.2.3：

Mass Concentration Methods / 5.3：

Construction of schemes / 5.3.1：

Error estimates for semi-discrete schemes / 5.3.2：

Error estimates for fully-discrete schemes / 5.3.3：

High Order Element Difference Schemes / 5.4：

Cubic element difference schemes for one-dimensional parabolic equations / 5.4.1：

Quadratic element difference schemes for two-dimensional parabolic equations / 5.4.2：

Generalized Difference Methods for Nonlinear Parabolic Equations / 5.5：

Hyperbolic Equations / 5.5.1：

Generalized Difference Methods for Second Order Hyperbolic Equations / 6.1：

Semi-discrete generalized difference scheme / 6.1.1：

Fully-discrete generalized difference scheme / 6.1.2：

Generalized Upwind Schemes for First Order Hyperbolic Equations / 6.2：

Generalized upwind schemes / 6.2.1：

Semi-discrete error estimates / 6.2.2：

Fully-discrete error estimates / 6.2.3：

Generalized Upwind Schemes for First Order Hyperbolic Systems / 6.3：

Integral forms / 6.3.1：

Generalized upwind difference schemes / 6.3.2：

Estimation of a bilinear form / 6.3.3：

Some practical difference schemes / 6.3.4：

A numerical example / 6.3.5：

Finite Volume Methods for Nonlinear Conservative Hyperbolic Equations / 6.4：

Convection-Dominated Diffusion Problems / 7：

One-Dimensional Characteristic Difference Schemes / 7.1：

Difference methods based on algebraic interpolations / 7.1.1：

Upwind difference schemes / 7.1.2：

Generalized Upwind Difference Schemes for Steady-state Problems / 7.2：

Construction of the difference schemes / 7.2.1：

Convergence and error estimate / 7.2.2：

Extreme value theorem and uniform convergence / 7.2.3：

Mass conservation / 7.2.4：

Generalized Upwind Difference Schemes for Nonsteady-state Problems / 7.3：

Construction of difference schemes / 7.3.1：

Highly Accurate Generalized Upwind Schemes / 7.3.2：

Upwind Schemes for Nonlinear Convection Problems / 7.4.1：

Applications / 8：

Planar Elastic Problems / 8.1：

Displacement methods / 8.1.1：

Mixed methods / 8.1.2：

Computation of Electromagnetic Fields / 8.2：

Numerical Simulation of Underground Water Pollution / 8.3：

Upwind weighted multi-element balancing method / 8.3.1：

Stokes Equation / 8.4：

Nonconforming generalized difference method / 8.4.1：

Coupled Sound-Heat Problems / 8.4.2：

Regularized Long Wave Equations / 8.6：

Semi-discrete generalized difference schemes / 8.6.1：

Fully-discrete generalized difference schemes / 8.6.2：

Hierarchical Basis Methods / 8.6.3：

Hierarchical Basis / 8.7.1：

Application to difference equations / 8.7.2：

Iteration methods / 8.7.3：

Bibliography / 8.7.4：

Index

Preface

Preliminaries / 1：

Sobolev Spaces / 1.1：

Smooth approximations. Fundamental lemma of variational methods / 1.1.1：

Generalized derivatives and Sobolev spaces / 1.1.2：

Imbedding and trace theorems / 1.1.3：

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