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
Contents Preface Chapter I. General Principle of the Natural Boundary Integral Method 1 1.1 Introduction 1 1.2 Boundary Reductions and Boundary Element Methods 5 1.2.1 Indirect boundary reduction 5 1.2.2 Direct boundary reduction 13 1.2.3 Numerical solutions of the boundary integral equations... 15 1.3 Basic Idea of the Natural Boundary Reduction 17 1.3.1 Natural boundary reduction of elliptic boundary value problems 18 1.3.2 Equivalent variational problems for the Neumann problem. 24 1.3.3 Expression for the natural integral operator 27 1.4 Numerical Computation of Hypersingular Integrals 29 1.4.1 The method using series expansion of the integral kernel... 30 1.4.2 The method of subtracting the singularity 33 1.4.3 Approximate integration formulas for finite-part integrals.. 40 1.4.4 The method of regularization and the indirect method 48 1.5 Convergence and Error Estimates for the Natural Boundary Element Method 50 1.5.1 Approximate variational problem and the convergence of its solution 51 1.5.2 Error estimates on the boundary 55 1.5.3 Error estimates in the domain 61 1.6 On Computation of Poisson Integral Formulas 63 1.6.1 The method of using special solutions 64 1.6.2 Error estimate 66 xiii
viii Content Chapter II. Boundary Value Problem for the Harmonic Equation 69 2.1 Introduction 69 2.2 Representation of a Solution by Complex Variable Functions.. 70 2.2.1 Theorem and its proof 71 2.2.2 Simple examples for application 72 2.3 Principle of the Natural Boundary Reduction 76 2.3.1 Variational problem in a domain 76 2.3.2 Natural boundary reduction and variational problem on boundary 79 2.4 Natural Integral Equations and Poisson Integral Formulas for Some Typical Domains 83 2.4.1 For the upper half-plane 84 2.4.2 For an interior circular domain 88 2.4.3 For an exterior circular domain 93 2.4.4 Some simple examples 94 2.5 Natural Boundary Reduction for General Simply Connected Domains 100 2.5.1 Conformal mapping and natural boundary reduction 100 2.5.2 Application to angular, sectorial, and rectangular domains 103 2.6 Natural Integral Operators and Their Inverse Operators 105 2.6.1 For the upper half plane 106 2.6.2 For interior and exterior circular domains 108 2.6.3 A theorem for general simply connected domains 113 2.7 Direct Study of Natural Integral Equations 119 2.7.1 For the upper half plane 119 2.7.2 For interior and exterior circular domains 120 2.8 Numerical Solution of Natural Integral Equations 125 2.8.1 Computation of coefficients of stiffness matrices 126 2.8.2 Condition number of stiffness matrices 134 2.8.3 Error estimates for the natural boundary element solution 137 2.8.4 Numerical examples 139 2.9 Numerical Solution of the Natural Integral Equation over a Sector with Crack or Concave Angle 145 2.9.1 Natural integral equation and its boundary element solution 145 2.9.2 Error estimates for the approximate solution 147
Content IX 2.9.3 Singularity of the solution 149 2.9.4 Numerical examples 151 Chapter III. Boundary Value Problem of the Biharmonic Equation 155 3.1 Introduction 155 3.2 Representation of a Solution by Complex Variable Functions. 158 3.2.1 Theorem and its proof 159 3.2.2 Simple examples for application 160 3.3 Principle of the Natural Boundary Reduction 164 3.3.1 Variational problem in domain 164 3.3.2 Natural boundary reduction and variational problem on boundary 170 3.4 Natural Integral Equations and Poisson Integral Formulas for Some Typical Domains 175 3.4.1 For the upper half-plane 175 3.4.2 For an interior circular domain 181 3.4.3 For an exterior circular domain 190 3.4.4 Some simple examples 197 3.5 Natural Integral Operators and Their Inverse Operators 201 3.5.1 For the upper half-plane 202 3.5.2 For an interior circular domain 203 3.5.3 For an exterior circular domain 206 3.6 Direct Study of Natural Integral Equations 207 3.6.1 For the upper half-plane 207 3.6.2 For an interior circular domain 209 3.6.3 For an exterior circular domain 216 3.7 Numerical Solution of Natural Integral Equations 222 3.7.1 Computation of coefficients of stiffness matrices 224 3.7.2 Error estimates for the natural boundary element solution 228 3.7.3 Numerical examples 231 3.8 Boundary Value Problems of Multi-Harmonic Equations 234 3.8.1 Representation of solution via complex variable functions. 235 3.8.2 Principle of the natural boundary reduction 237 3.8.3 Some results for the upper half-plane 239 Chapter IV. Plane Elasticity Problem 245 4.1 Introduction 245
X Content 4.2 Representation of a Solution by Complex Variable Functions. 248 4.2.1 Theorem and its proof 249 4.2.2 Simple examples for application 252 4.3 Principle of the Natural Boundary Reduction 257 4.3.1 Variational problem in domain 257 4.3.2 Natural boundary reduction and variational problem on boundary 260 4.4 Natural Integral Equations and Poisson Integral Formulas for Some Typical Domains 263 4.4.1 For the upper half-plane 264 4.4.2 For an interior circular domain 269 4.4.3 For an exterior circular domain 280 4.4.4 Some simple examples 286 4.5 Natural Integral Operators and Their Inverse Operators 291 4.5.1 For the upper half-plane 291 4.5.2 For an interior circular domain 293 4.5.3 For an exterior circular domain 301 4.6 Direct Study of Natural Integral Equations 303 4.6.1 For the upper half-plane 304 4.6.2 For an interior circular domain 305 4.6.3 For an exterior circular domain 320 4.7 Numerical Solution of Natural Integral Equations 322 4.7.1 Computation of coefficients of stiffness matrices 324 4.7.2 Error estimates for the natural boundary element solution 329 4.7.3 Numerical examples 330 Chapter V. Stokes' Problem 333 5.1 Introduction 333 5.2 Representation of a Solution by Complex Variable Functions. 334 5.2.1 Theorem and its proof 334 5.2.2 Simple examples for application 338 5.3 Principle of the Natural Boundary Reduction 342 5.3.1 Green's formula 343 5.3.2 Natural boundary reduction and equivalent variational problems 346 5.4 Natural Integral Equations and Poisson Integral Formulas for Some Typical Domains 349
Content XI 5.4.1 For the upper half-plane 349 5.4.2 For an exterior circular domain 353 5.4.3 For an interior circular domain 361 5.4.4 Some simple examples 364 5.5 Natural Integral Operators and Their Inverse Operators 369 5.5.1 For the upper half-plane 370 5.5.2 For an exterior circular domain 371 5.5.3 For an interior circular domain 377 5.6 Direct Study of Natural Integral Equations 384 5.6.1 For the upper half-plane 384 5.6.2 For an exterior circular domain 385 5.6.3 For an interior circular domain 394 5.7 Numerical Solution of Natural Integral Equations 406 5.7.1 Computation of coefficients of stiffness matrices 407 5.7.2 Error estimates for the natural boundary element solution 411 5.7.3 Numerical example 412 Chapter VI. The Coupling of Natural Boundary Elements and Finite Elements 415 6.1 Introduction 415 6.2 The Coupling for the Harmonic Boundary Value Problem 417 6.2.1 For a cracked domain 417 6.2.2 For an unbounded domain 426 6.2.3 Numerical example 430 6.3 The Coupling for the Biharmonic Boundary Value Problem.. 434 6.3.1 Principle of the coupling 434 6.3.2 Convergence and error estimates 436 6.4 The Coupling for the Plane Elasticity Problem 440 6.4.1 Principle of the coupling 440 6.4.2 Convergence and error estimates 442 6.5 The Coupling for Stokes' Problem 444 6.5.1 Principle of the coupling 444 6.5.2 Convergence and error estimates 448 6.6 Approximation of Boundary Conditions at Infinity 449 6.6.1 Approximation of integral boundary conditions 450 6.6.2 Error estimates 455
xii Content Chapter VII. Domain Decomposition Methods Based On Natural Boundary Reduction 465 7.1 Introduction 465 7.2 Overlapping DDM based on natural boundary reduction 467 7.2.1 Schwarz' alternating method and its convergence 469 7.2.2 Analysis of convergence rate 473 7.2.3 Some examples 476 7.3 Non-overlapping DDM based on natural boundary reduction. 479 7.3.1 Dirichlet-Neumann alternating method and its convergence 479 7.3.2 Discretization and its convergence 487 7.3.3 Numerical examples 492 7.3.4 Application to plane elasticity problem 498 7.4 Steklov-Poincare operators and their inverse operators 507 7.4.1 Elliptic problem of second order 507 7.4.2 Biharmonic problem 510 7.4.3 Plane elasticity problem 514 7.4.4 Stokes' problem 516 7.4.5 Preconditioning the Steklov-Poincare operators 519 References 521 Index 535