Natural Boundary Integral Method and Its Applications
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1 Natural Boundary Integral Method and Its Applications By De-hao Yu State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing, People's Republic of China Science Press Beijing/New York, Kluwer Academic Publishers Dordrecht/Boston/London
2 Contents Preface Chapter I. General Principle of the Natural Boundary Integral Method Introduction Boundary Reductions and Boundary Element Methods Indirect boundary reduction Direct boundary reduction Numerical solutions of the boundary integral equations Basic Idea of the Natural Boundary Reduction Natural boundary reduction of elliptic boundary value problems Equivalent variational problems for the Neumann problem Expression for the natural integral operator Numerical Computation of Hypersingular Integrals The method using series expansion of the integral kernel The method of subtracting the singularity Approximate integration formulas for finite-part integrals The method of regularization and the indirect method Convergence and Error Estimates for the Natural Boundary Element Method Approximate variational problem and the convergence of its solution Error estimates on the boundary Error estimates in the domain On Computation of Poisson Integral Formulas The method of using special solutions Error estimate 66 xiii
3 viii Content Chapter II. Boundary Value Problem for the Harmonic Equation Introduction Representation of a Solution by Complex Variable Functions Theorem and its proof Simple examples for application Principle of the Natural Boundary Reduction Variational problem in a domain Natural boundary reduction and variational problem on boundary Natural Integral Equations and Poisson Integral Formulas for Some Typical Domains For the upper half-plane For an interior circular domain For an exterior circular domain Some simple examples Natural Boundary Reduction for General Simply Connected Domains Conformal mapping and natural boundary reduction Application to angular, sectorial, and rectangular domains Natural Integral Operators and Their Inverse Operators For the upper half plane For interior and exterior circular domains A theorem for general simply connected domains Direct Study of Natural Integral Equations For the upper half plane For interior and exterior circular domains Numerical Solution of Natural Integral Equations Computation of coefficients of stiffness matrices Condition number of stiffness matrices Error estimates for the natural boundary element solution Numerical examples Numerical Solution of the Natural Integral Equation over a Sector with Crack or Concave Angle Natural integral equation and its boundary element solution Error estimates for the approximate solution 147
4 Content IX Singularity of the solution Numerical examples 151 Chapter III. Boundary Value Problem of the Biharmonic Equation Introduction Representation of a Solution by Complex Variable Functions Theorem and its proof Simple examples for application Principle of the Natural Boundary Reduction Variational problem in domain Natural boundary reduction and variational problem on boundary Natural Integral Equations and Poisson Integral Formulas for Some Typical Domains For the upper half-plane For an interior circular domain For an exterior circular domain Some simple examples Natural Integral Operators and Their Inverse Operators For the upper half-plane For an interior circular domain For an exterior circular domain Direct Study of Natural Integral Equations For the upper half-plane For an interior circular domain For an exterior circular domain Numerical Solution of Natural Integral Equations Computation of coefficients of stiffness matrices Error estimates for the natural boundary element solution Numerical examples Boundary Value Problems of Multi-Harmonic Equations Representation of solution via complex variable functions Principle of the natural boundary reduction Some results for the upper half-plane 239 Chapter IV. Plane Elasticity Problem Introduction 245
5 X Content 4.2 Representation of a Solution by Complex Variable Functions Theorem and its proof Simple examples for application Principle of the Natural Boundary Reduction Variational problem in domain Natural boundary reduction and variational problem on boundary Natural Integral Equations and Poisson Integral Formulas for Some Typical Domains For the upper half-plane For an interior circular domain For an exterior circular domain Some simple examples Natural Integral Operators and Their Inverse Operators For the upper half-plane For an interior circular domain For an exterior circular domain Direct Study of Natural Integral Equations For the upper half-plane For an interior circular domain For an exterior circular domain Numerical Solution of Natural Integral Equations Computation of coefficients of stiffness matrices Error estimates for the natural boundary element solution Numerical examples 330 Chapter V. Stokes' Problem Introduction Representation of a Solution by Complex Variable Functions Theorem and its proof Simple examples for application Principle of the Natural Boundary Reduction Green's formula Natural boundary reduction and equivalent variational problems Natural Integral Equations and Poisson Integral Formulas for Some Typical Domains 349
6 Content XI For the upper half-plane For an exterior circular domain For an interior circular domain Some simple examples Natural Integral Operators and Their Inverse Operators For the upper half-plane For an exterior circular domain For an interior circular domain Direct Study of Natural Integral Equations For the upper half-plane For an exterior circular domain For an interior circular domain Numerical Solution of Natural Integral Equations Computation of coefficients of stiffness matrices Error estimates for the natural boundary element solution Numerical example 412 Chapter VI. The Coupling of Natural Boundary Elements and Finite Elements Introduction The Coupling for the Harmonic Boundary Value Problem For a cracked domain For an unbounded domain Numerical example The Coupling for the Biharmonic Boundary Value Problem Principle of the coupling Convergence and error estimates The Coupling for the Plane Elasticity Problem Principle of the coupling Convergence and error estimates The Coupling for Stokes' Problem Principle of the coupling Convergence and error estimates Approximation of Boundary Conditions at Infinity Approximation of integral boundary conditions Error estimates 455
7 xii Content Chapter VII. Domain Decomposition Methods Based On Natural Boundary Reduction Introduction Overlapping DDM based on natural boundary reduction Schwarz' alternating method and its convergence Analysis of convergence rate Some examples Non-overlapping DDM based on natural boundary reduction Dirichlet-Neumann alternating method and its convergence Discretization and its convergence Numerical examples Application to plane elasticity problem Steklov-Poincare operators and their inverse operators Elliptic problem of second order Biharmonic problem Plane elasticity problem Stokes' problem Preconditioning the Steklov-Poincare operators 519 References 521 Index 535
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