Integral EquationsTheory and Numerical Treatment

Wolfgang Hackbusch

6Forschungs- und Technoiogia2QP4rum

Bibiiothsk

n an DSt:

31Sam:

Birkhauser VerlagBasel Boston Berlin

Contents

Preface , v

Contents vii

Notations xiii

1 Introduction 11.1 Integral Equations 11.2 Basics from Analysis 3

1.2.1 Continuous Functions 31.2.2 Lipschitz Continuous Functions 31.2.3 Holder Continuous Functions 4

1.3 Basics from Functional Analysis S1.3.1 Banach Spaces 51.3.2 Banach Spaces Cx(D), c£(D), Cx(D) 61.3.3 Banach Spaces LUD), L2(D),LCO(D) 71.3.4 Dense Subspaces 81.3.5 Banach's Fixed Point Theorem 81.3.6 Linear Operators 91.3.7 Theorem of Uniform Boundedness 101.3.8 Compact Sets and Compact Mappings 111.3.9 Riesz-Schauder Theory 141.3.10 Hilbert Spaces, Orthogonal Complements, Projections . . . . 14

1.4 Basics from Numerical Mathematics IS1.4.1 Interpolation 151.4.2 Quadrature 191.4.3 Condition Number of a System of Equations 23

2 Volterra Integral Equations 252.1 Theory of Volterra Integral Equations of the Second Kind . . . . 25

2.1.1 Existence und Uniqueness of the Solution 252.1.2 Regularity of the Solution 27

2.2 Numerical Solution by Quadrature Methods 292.2.1 Derivation of the Discretisation 292.2.2 Error Estimate 30

2.3 Further Numerical Methods 372.4 Linear Volterra Integral Equations of Convolution Type . . . . 392.5 The Volterra Integral Equations of the First Kind 41

Contents ix

4.4.5 Numerical Examples 884.5 Galerkin Method 91

4.5.1 Subspace, Orthogonal Projection 914.5.2 Derivation of the System of Equations 924.5.3 Convergence in L2(D) and L^tD) 934.5.4 Error Estimates 964.5.5 Condition Number of the System of Equations 974.5.6 Example: Piecewise Constant Functions 1004.5.7 Example: Piecewise Linear Functions 1044.5.8 General Analysis of Projection Errors 1054.5.9 Revisited: Piecewise Linear Functions 1074.5.10 Numerical Examples 109

4.6 Additional Comments Concerning Projection Methods I l l4.6.1 Regularisation Method . . . : I l l4.6.2 Estimates with Respect to Weaker Norms 1124.6.3 The Iterated Approximation 1164.6.4 Superconvergence 1184.6.5 More General Formulations of the Projection Method . . . . 1214.6.6 Numerical Quadrature 1234.6.7 Product Integration 126

4.7 Discretisation by Quadrature: The Nystrom Method 1284.7.1 Description of the Method 1284.7.2 Convergence Analysis 1304.7.3 Stability 1324.7.4 Consistency Order 1364.7.5 Condition Number of the System of Equations 1374.7.6 Regularisation 1384.7.7 Numerical Examples 1394.7.8 Product Integration 139

4.8 Supplements 1414.8.1 Connection between the Discretisation Methods 141

4.8.1.1 The Kernel Approximation and the Galerkin Method . . . 1414.8.1.2 From the Galerkin Method to the Collocation and

Nystrom-Method 1414.8.1.3 From the Collocation to the Nystrom Method 1424.8.1.4 From the Collocation to the Galerkin Method 143

4.8.2 Method of the Defect Correction 1434.8.3 Extrapolation Method 1444.8.4 Eigenvalue Problems 1474.8.5 Complementary Integral Equations 1524.8.6 Supplement: Perturbation Theorem for Stability . . . . . . . 154

5 Multi-Grid Methods for Solving Systems Arising from IntegralEquations of the Second Kind 1555.1 Preliminaries 155

5.1.1 Notation 1555.1.2 Direct Solution of the System of Equations 1565.1.3 Picard Iteration 1565.1.4 Conjugate Gradient Method 158

5.2 Stability and Convergence (Discrete Formulation) 1605.2.1 Prolongations and Restrictions 1605.2.2 The Banach Space Y and the Discrete Spaces yn 163

Contents

5.2.3 The Interpolation Error or Projection Error 1655.2.4 Consistency 1655.2.5 Stability 1665.2.6 Convergence 167

5.3 The Hierarchy of Discrete Problems 1685.3.1 Levels of Discretisations 1685.3.2 Prolongations and Restrictions 1685.3.3 Relative Consistency 1725.3.4 Convergence 173

5.4 Two-Grid Iteration 1745.4.1 The Two-Grid Algorithm 1745.4.2 Convergence Analysis 1755.4.3 Amount of Computational Work 1765.4.4 Variant for A,,*/ 1785.4.5 Numerical Examples 179

5.5 Multi-Grid Iteration 1815.5.1 Algorithm (Basic Version) 1815.5.2 Amount of Computational Work . 1835.5.3 Convergence 1845.5.4 Numerical Examples 1885.5.5 Variants of the Multi-Grid Methods 190

5.6 Nested Iteration 1955.6.1 Algorithm 1955.6.2 Amount of Computational Work 1965.6.3 Convergence 1975.6.4 Numerical Examples 1995.6.5 Nested Iteration with Nystrom Interpolation 199

6 Abel's Integral Equation 2016.1 Notations and Examples 201

6.1.1 Abel's Integral Equation and its Generalisations 2016.1.2 Examples from Applications 2016.1.3 Improper Integrals 203

6.2 A Necessary Condition for a Bounded Solution 2066.3 Euler's Integrals 2076.4 Inversion of Abel's Integral Equation 2086.5 Reformulation for Kernels k(x,y)/(x-y)x 2136.6 Numerical Methods for Abel's Integral Equation 214

7 Singular Integral Equations 2167.1 The Cauchy Principal Value 216

7.1.1 Definition and Properties 2167.1.2 Curvilinear Integrals 2207.1.3 Cauchy's Principal Value for Curvilinear Integrals 2227.1.4 The Example f(X,) = l/(l,-z) 224

7.2 The Cauchy Kernel 2307.2.1 Definition and Properties 2307.2.2 Regularity Properties 2347.2.3 Properties of the Generated Holomorphic Function 2367.2.4 Representation of K2 244

Contents ?ri

7.2.5 The Cauchy Integral on the Unit Circle 2457.3 The Singular Integral Equation 247

7.3.1 The Case of Cons t an t Coefficients 2477.3.2 The Case of Variable Coefficients 2477.3.3 General Singular Integral Equat ions 2487.3.4 Approximat ion of the Cauchy Integra l on the Uni t Circle . . . 2497.3.5 Approximat ion of the Cauchy Integral on an Arbi t rary Curve F 2507.3.6 Mult i -Grid Methods for Equat ions of a Special Form . . . . 251

7.4 Application to the Dirichlet Problem for Laplace's Equat ion . . . 2537.4.1 The Problem in the Inter ior Domain 2537.4.2 The Double-Layer Potent ia l 2537.4.3 Uniqueness and Representa t ion Theorem 2577.4.4 The Case of a Smooth Boundary f 2597.4.5 The Double-Layer Potent ia l for Solving the Exter ior Problem . 2607.4.6 The Tangential Derivative of the Single-Layer Potent ia l . . . 262

7.5 Hypersingular Integrals 264

8 The Integral Equation Method 2668.1 The Single-Layer Potential 266

8.1.1 The Singularity Function 2668.1.2 Continuity of the Single-Layer Potential 268

8.1.2.1 Definition 2688.1.2.2 Surface Integrals 2688.1.2.3 Improper Integrals on Surfaces 2698.1.2.4 Properties of the Single-Layer Potential 271

8.1.3 Derivatives of the Single-Layer Potential 2718.1.3.1 The Normal Derivative 2718.1.3.2 The Cauchy Principal Value for Surface Integrals 2758.1.3.3 Other Directional Derivatives 279

8.1.4 Formulation of the Dirichlet Boundary Value Problem as FirstKind Integral Equation for the Single-Layer Potential . . . . 281

8.1.4.1 Concerning the Interior and Exterior Problem of theLaplace Equation 281

8.1.4.2 The Integral Equation of the First Kind 2838.1.5 Formulation of the Neumann Boundary Value Problem as

Second Kind Integral Equation for the Single-Layer Potential . 2848.2 The Double-Layer Potential 287

8.2.1 Definition 2878.2.2 Regularity Properties of the Double-Layer Integral Operator . 2888.2.3 Jump Properties of the Double-Layer Potential 2928.2.4 Further Properties of the Double-Layer Potential 294

8.2.4.1 Holder Continuity 2948.2.4.2 ' The Potential close to a Jump Discontinuity of the Density 2958.2.4.3 The Double-Layer Potential of the Density f = l 296

8.2.5 Derivatives of the Double-Layer Potential 2998.2.6 Integral Equations with the Double-Layer Operator 303

8.2.6.1 Formulation of the Dirichlet Boundary Value Problem asIntegral Equation of the Second Kind with the Double-Layer Operator 303

8.2.6.2 Formulation of the Neumann Boundary Value Problem asIntegral Equation of the Second Kind with the Double-Layer Operator 305

xii Contents

8.2.7 Non-smooth Curves or Surfaces 3068.3 The Hypersingular Integral Equation 3098.4 Synopsis: Integral Equations for the Laplace Equation 3138.5 The Integral Equation Method for Other Differential Equations . 314

8.5.1 Differential Equations of Second Order 3148.5.2 Equations of Higher Order 3158.5.3 Systems of Differential Equations 316

9 The Boundary Element Method 3189.1 Construction of the Boundary Element Method 318

9.1.1 Definition of the Boundary Element Method 3189.1.2 Galerkin Method 3189.1.3 Collocation Method 3199.1.4 Convergence in the Compact Case 3209.1.5 Convergence in the Case of Elliptic Bilinear-Forms 320

9.2 The Boundary Elements 3239.2.1 Elements in the Two-Dimensional Case 3239.2.2 Geometric Discretisation 3249.2.3 Elements in the Three-Dimensional Case 3259.2.4 Error Considerations 326

9.3 Multi-Grid Methods 3279.3.1 Equations of the Second Kind 3279.3.2 Equations of the First Kind 328

9.4 Integration and Numerical Quadrature 3309.4.1 General Considerations 3309.4.2 Weakly Singular Integrals 3309.4.3 Nearly Singular Integrals 3309.4.4 Strongly Singular Integrals 3339.4.5 Treatment of Double Integrals Arising from the Galerkin

Method 3339.5 Solution of Inhomogeneous Equations 3369.6 Computation of the Potential 336

9.6.1 Evaluation of the Potential 3369.6.2 Evaluation of the Derivatives 3369.6.3 Error Considerations 3369.6.4 Extrapolation • 337

9.7 The Panel Clustering Algorithm 3379.7.1 Introduction 3379.7.2 Panels 3389.7.3 The Panel Clustering Method 338

9.7.3.1 The Far Field Expansion 3389.7.3.2 Panel Clustering 3389.7.3.3 Admissible Clusters and Admissible Coverings 3399.7.3.4 The Algorithm for Matrix Multiplication 340

9.7.4 The Additional Quadrature Error 3429.7.5 Complexity of the Algorithm 342

Bibliography 344

Index 353