SUB Gttttingen 7 208 427 244 98 A 5141 Discrete Projection Methods for Integral Equations M.A. Golberg & C.S. Chen TM Computational Mechanics Publications Southampton UK and Boston USA
Contents Sources of Integral Equations 1 1.1 Introduction 1 1.2 Initial value problems 1 1.3 Boundary value problems for ordinary differential equations 4 1.4 Integral equations for partial differential equations 7 1.4.1 The direct method 10 1.5 The Abel equation 11 1.6 References 12 Classification of Integral Equations 15 2.1 Introduction 15 2.2 Linear equations 15 2.2.1 Kind and type 15 2.2.2 Kernel classification 17 2.3 Nonlinear equations 18 2.4 Special equations 19 2.4.1 Abel equations 19 2.4.2 Cauchy singular equations 20 2.4.3 Hypersingular equations 21 2.5 Boundary integral equations 21 2.6 References 22 Some Analytic Methods for Solving Integral Equations 25 3.1 Introduction 25 3.2 The Abel equation 26 3.2.1 The generalized Abel equation 27 3.3 Volterra equations of the second kind 29 3.3.1 Convolution kernels 29 3.3.2 Degenerate kernels 31 3.3.3 Continuous kernels. 32 3.4 FVedholm equations of the second kind 34 3.4.1 Degenerate kernels 34 3.4.2 Semidegenerate kernels 36 3.4.3 The boundary value method 37
3.4.4 Continuous kernels 39 3.5 FYedholm equations of the first kind 40 3.6 The airfoil equation 42 3.7 References 47 Functional Analysis 51 4.1 Introduction 51 4.2 Vector spaces 52 4.3 Subspaces 53 4.4 Linear combinations 55 4.5 Spanning sets, bases and dimension 55 4.6 Linear transformations 57 4.6.1 Properties of linear transformations 58 4.6.2 Powers of T 59 4.6.3 Inverses 59 4.7 Normed vector spaces 61 4.7.1 Properties of inner product spaces 63 4.8 Metric spaces and convergence 64 4.8.1 Metric convergence 65 4.8.2 Cauchy sequences and completeness 67 4.8.3 Equivalent norms 69 4.9 The contraction mapping theorem 71 4.9.1 Some applications of the CMT 73 4.10 Bounded linear operators 75 4.11 The dual space 78 4.12 Banach's lemma 81 4.12.1 Split equations of the first kind 82 4.13 The Predholm alternative 83 4.13.1 Hilbert-Schmidt operators 87 4.14 Approximation theory 90 4.14.1 Projection methods 94 4.14.2 Projection methods for Hu = Ku + f 99 4.14.3 Iterated projection methods 101 4.15 Superconvergence of linear functionals 104 4.16 Perturbed projection methods 105 4.17 Brakhage's lemma 107 4.18 Further variants of projection methods 109 4.19 Conditioning 110 4.19.1 Stability of Galerkin's method 112 4.19.2 Stability of projection methods 114 4.20 References 115
Approximation and Numerical Integration 117 5.1 Introduction 117 5.2 One dimensional approximation 118 5.2.1 -Taylor polynomials 118 5.2.2 Polynomial interpolation 119 5.2.3 Hermite interpolation 123 5.2.4 Piecewise polynomial interpolation 124 5.2.5 Trigonometric interpolation 125 5.2.6 Other approximation methods 126 5.3 Multivariate approximation 129 5.3.1 Multivariate interpolation 130 5.3.2 Radial basis function interpolation 132 5.3.3 Approximation on the unit sphere in R 3 134 5.4 Numerical integration 136 5.4.1 Taylor polynomial rules 136 5.4.2 Interpolatory rules 137 5.4.3 Error analysis for interpolatory rules 139 5.4.4 Compound rules 142 5.4.5 Gaussian integration 144 5.5 Orthogonal interpolation 151 5.6 Numerical approximation of Fourier coefficients 153 5.7 Numerical approximation of integral transforms 157 5.8 Cauchy singular integrals 161 5.9 Numerical approximation of multiple integrals 163 5.9.1 Numerical integration on triangles 167 5.9.2 Numerical integration on the sphere 169 5.10 References 170 Discrete Projection Methods for Fredholm Equations 175 6.1 Introduction 175 6.2 Projection methods for one-dimensional equations 177 6.2.1 Galerkin's method - orthogonal polynomial bases 177 6.2.2 The discrete Galerkin method 178 6.2.3 Trigonometric approximations 183 6.2.4 Piecewise polynomial approximation 186 6.2.5 Collocation 189 6.2.6 Orthogonal collocation 190 6.2.7 Discrete orthogonal collocation 192 6.2.8 Trigonometric collocation 193 6.2.9 Piecewise polynomial collocation 195 6.2.10 Superconvergence of collocation for Volterra equations 199 6.2.11 Multiquadric collocation 200 6.2.12 Stability of collocation.202 6.3 Quadrature methods 204
6.3.1 The Nystrom method: continuous kernels 205 6.3.2 Product quadrature 211 6.3.3 Stability of the Nystrom method 213 6.4 Multivariable equations 214 6.4.1 Galerkin's method 214 6.4.2 Direct analysis of the discrete Galerkin method 221 6.4.3 Piecewise polynomial collocation 225 6.4.4 Direct analysis of the discrete collocation method 228 6.4.5 Rbf collocation... 230 6.5 Wiener-Hopf equations 231 6.5.1 Piecewise polynomial collocation 233 6.5.2 Extensions to CJ 1 ".237 6.6 Nonlinear Equations 237 6.6.1 Iterated projection methods 240 6.6.2 Discrete projection methods 242 6.6.3 Hammerstein equations - the method of Kumar and Sloan... 243 6.7 References 246 Discrete Projection Methods for Cauchy Singular Equations 251 7.1 Introduction 251 7.2 Numerical methods for the GAE 252 7.2.1 Operator formulation of the GAE 252 7.2.2 Degenerate kernel methods 255 7.2.3 Galerkin's method: v = 0 257 7.2.4 The Sloan iterate 260 7.2.5 Logarithmically singular kernels 260 7.2.6 The flap problem 262 7.2.7 Discrete Galerkin methods: v = 0 265 7.2.8 Galerkin's method: v = 1 266 7.2.9 ' Collocation v = 0: continuous data 270 7.2.10 Discrete collocation methods: u = 0 274 7.2.11 Collocation: v = ±1 274 7.2.12 Quadrature methods 275 7.3 CSIEs of the second kind with constant coefficients 279 7.3.1 The standard polynomial algorithms 280 7.3.2 Galerkin's method 283 7.3.3 Collocation 284 7.3.4 Quadrature methods 284 7.4 Further convergence results 286 7.4.1 Galerkin's method: v = 0 286 7.4.2 The discrete Galerkin method: v = 0 287 7.4.3 Galerkin's method: v = 1 289 7.4.4 Discrete Galekin's method: v = 1 290 7.4.5 Collocation: v = 0 291
7.4.6 Discrete collocation: v = 0 291 7.4.7 Discrete collocation: v \ 293 7.4.8 Chebyshev collocation 293 7.5 Stability 297 7.6 Convergence of collocation with logarimithcally singular kernels... 299 7.7 Convergence of the Lagrange interpolant of the Gaussian quadrature method 301 7.8 The discrete Sloan iterate 303 7.9 Hypersingular equations 305 7.10 References 309 Boundary Integral Equations in R 2 315 8.1 Introduction 315 8.2 Partial differential equations equivalent to Laplace's equation 316 8.2.1 Poisson's equation 316 8.2.2 Rbf approximations for particular solutions 319 8.2.3 Semilinear equations 323 8.2.4 The Kirchhoff transformation 324 8.3 Boundary integral equation solution of Laplace's equation-smooth domains 325 8.3.1 The Dirichlet problem - double layer potential 325 8.3.2 The Dirichlet problem - single layer potential 328 8.3.3 Galerkin's method 330 8.3.4 A quadrature method 333 8.3.5 The delta-trigonometric method 336 8.4 Other boundary value problems for Laplace's equation 340 8.5 The method of fundamental solutions 342 8.5.1 MFS for Poisson's equation 346 8.5.2 A thermal explosion problem 349 8.5.3 Further aspects of the MFS 350 8.6 Boundaries with corners 354 8.7 Cea's lemma 359 8.8 References 364 Boundary Integral Equations in R 3 369 9.1 Introduction 369 9.2 Integral equations on the sphere 370 9.2.1 A Galerkin method 371 9.2.2 A discrete Galerkin method 373 9.2.3 A Nystrom method 375 9.2.4 A collocation method 376 9.3 The BEM for boundary integral equations 377 9.3.1 Piecewise quadratic approximation 380 9.3.2 Discrete collocation methods 384
9.3.3 The approximate surface 385 9.4 Boundary value problems for Laplace's equation 386 9.5 Time dependent problems and the Helmholtz equation 390 9.6 The Dirichlet problem - double layer potential 391 9.6.1 Spherical harmonic approximation 393 9.6.2 The single layer potential 395 9.6.3 Boundary element approximation 398 9.7 Boundary integral equation solution of the Helmholtz equation 400 9.8 The method of fundamental solutions 402 9.8.1 Laplace's equation 402 9.8.2 MFS for Poisson's equation 405 9.9 References 406