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Elliptic Differential Equations : Theory and Numerical Treatment - Wolfgang Hackbusch

Elliptic Differential Equations

Theory and Numerical Treatment

By: Wolfgang Hackbusch, R. Fadiman (Translator), P.D.F. Ion (Translator)

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This book has developed from lectures that the author gave for mathematics students at the Ruhr-Universitat Bochum and the Christian-Albrechts-Uni­ versitat Kiel. This edition is the result of the translation and correction of the German edition entitled Theone und Numenk elliptischer Differential­ gleichungen. The present work is restricted to the theory of partial differential equa­ tions of elliptic type, which otherwise tends to be given a treatment which is either too superficial or too extensive. The following sketch shows what the problems are for elliptic differential equations. A: Theory of B: Discretisation: c: Numerical analysis elliptic Difference Methods, convergence, equations finite elements, etc. stability Elliptic Discrete boundary value equations f-------- ----- problems E:Theory of D: Equation solution: iteration Direct or with methods iteration methods The theory of elliptic differential equations (A) is concerned with ques­ tions of existence, uniqueness, and properties of solutions. The first problem of VI Foreword numerical treatment is the description of the discretisation procedures (B), which give finite-dimensional equations for approximations to the solu­ tions. The subsequent second part of the numerical treatment is numerical analysis (0) of the procedure in question. In particular it is necessary to find out if, and how fast, the approximation converges to the exact solution.

Table of Contents
Partial Differential Equations and Their Classification Into Typesp. 1
Examplesp. 1
Classification of Second-Order Equations into Typesp. 4
Type Classification for Systems of First Orderp. 6
Characteristic Properties of the Different Typesp. 7
The Potential Equationp. 12
Posing the Problemp. 12
Singularity Functionp. 14
The Mean Value Property and Maximum Principlep. 17
Continuous Dependence on the Boundary Datap. 23
The Poisson Equationp. 27
Posing the Problemp. 27
Representation of the Solution by the Green Functionp. 28
The Green Function for the Ballp. 34
The Neumann Boundary Value Problemp. 35
The Integral Equation Methodp. 36
Difference Methods for the Poisson Equationp. 38
Introduction: The One-Dimensional Casep. 38
The Five-Point Formulap. 40
M-matrices, Matrix Norms, Positive Definite Matricesp. 44
Properties of the Matrix L[subscript h]p. 53
Convergencep. 59
Discretisations of Higher Orderp. 62
The Discretisation of the Neumann Boundary Value Problemp. 65
Discretisation in an Arbitrary Domainp. 78
General Boundary Value Problemsp. 85
Dirichlet Boundary Value Problems for Linear Differential Equationsp. 85
General Boundary Conditionsp. 95
Boundary Problems of Higher Orderp. 103
Tools from Functional Analysisp. 110
Banach Spaces and Hilbert Spacesp. 110
Sobolev Spacesp. 115
Dual Spacesp. 130
Compact Operatorsp. 135
Bilinear Formsp. 137
Variational Formulationp. 144
Historical Remarksp. 144
Equations with Homogeneous Dirichlet Boundary Conditionsp. 145
Inhomogeneous Dirichlet Boundary Conditionsp. 150
Natural Boundary Conditionsp. 152
The Method of Finite Elementsp. 161
The Ritz-Galerkin Methodp. 161
Error Estimatesp. 167
Finite Elementsp. 171
Error Estimates for Finite Element Methodsp. 185
Generalisationsp. 193
Finite Elements for Non-Polygonal Regionsp. 196
Additional Remarksp. 199
Properties of the Stiffness Matrixp. 203
Regularityp. 208
Solutions of the Boundary Value Problem in [actual symbol not reproducible]p. 208
Regularity Properties of Differences Equationsp. 226
Special Differential Equationsp. 244
Differential Equations with Discontinuous Coefficientsp. 244
A Singular Perturbation Problemp. 247
Eigenvalue Problemsp. 253
Formulation of Eigenvalue Problemsp. 253
Finite Element Discretisationp. 254
Discretisation by Difference Methodsp. 267
Stokes Equationsp. 275
Systems of Elliptic Differential Equationsp. 275
Variational Formulationp. 278
Mixed Finite-Element Method for the Stokes Problemp. 290
Bibliographyp. 300
Indexp. 307
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540548225
ISBN-10: 354054822X
Series: Springer Series in Computational Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 311
Published: 13th June 2003
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 1.91
Weight (kg): 0.64