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# 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.

 Foreword Table of Contents Notation Partial Differential Equations and Their Classification Into Types p. 1 Examples p. 1 Classification of Second-Order Equations into Types p. 4 Type Classification for Systems of First Order p. 6 Characteristic Properties of the Different Types p. 7 The Potential Equation p. 12 Posing the Problem p. 12 Singularity Function p. 14 The Mean Value Property and Maximum Principle p. 17 Continuous Dependence on the Boundary Data p. 23 The Poisson Equation p. 27 Posing the Problem p. 27 Representation of the Solution by the Green Function p. 28 The Green Function for the Ball p. 34 The Neumann Boundary Value Problem p. 35 The Integral Equation Method p. 36 Difference Methods for the Poisson Equation p. 38 Introduction: The One-Dimensional Case p. 38 The Five-Point Formula p. 40 M-matrices, Matrix Norms, Positive Definite Matrices p. 44 Properties of the Matrix L[subscript h] p. 53 Convergence p. 59 Discretisations of Higher Order p. 62 The Discretisation of the Neumann Boundary Value Problem p. 65 Discretisation in an Arbitrary Domain p. 78 General Boundary Value Problems p. 85 Dirichlet Boundary Value Problems for Linear Differential Equations p. 85 General Boundary Conditions p. 95 Boundary Problems of Higher Order p. 103 Tools from Functional Analysis p. 110 Banach Spaces and Hilbert Spaces p. 110 Sobolev Spaces p. 115 Dual Spaces p. 130 Compact Operators p. 135 Bilinear Forms p. 137 Variational Formulation p. 144 Historical Remarks p. 144 Equations with Homogeneous Dirichlet Boundary Conditions p. 145 Inhomogeneous Dirichlet Boundary Conditions p. 150 Natural Boundary Conditions p. 152 The Method of Finite Elements p. 161 The Ritz-Galerkin Method p. 161 Error Estimates p. 167 Finite Elements p. 171 Error Estimates for Finite Element Methods p. 185 Generalisations p. 193 Finite Elements for Non-Polygonal Regions p. 196 Additional Remarks p. 199 Properties of the Stiffness Matrix p. 203 Regularity p. 208 Solutions of the Boundary Value Problem in [actual symbol not reproducible] p. 208 Regularity Properties of Differences Equations p. 226 Special Differential Equations p. 244 Differential Equations with Discontinuous Coefficients p. 244 A Singular Perturbation Problem p. 247 Eigenvalue Problems p. 253 Formulation of Eigenvalue Problems p. 253 Finite Element Discretisation p. 254 Discretisation by Difference Methods p. 267 Stokes Equations p. 275 Systems of Elliptic Differential Equations p. 275 Variational Formulation p. 278 Mixed Finite-Element Method for the Stokes Problem p. 290 Bibliography p. 300 Index p. 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