A convergence theorem for a class of Nyström methods for weakly singular integral equations on surfaces in R3

A convergence theorem is proved for a class of Nystrom methods for weakly singular integral equations on surfaces in three dimensions. Fredholm equations of the second kind as arise in connection with linear elliptic boundary value problems for scalar and vector fields are considered. In contrast to methods based on product integration, coordinate transformations and singularity subtraction, the family of Nystrom methods studied here is based on a local polynomial correction determined by an auxiliary system of moment equations. The polynomial correction is shown to remove the weak singularity in the integral equation and provide control over the approximation error. Convergence results for the family of methods are established under minimal regularity assumptions consistent with classic potential theory. Rates of convergence are shown to depend on the regularity of the problem, the degree of the polynomial correction, and the order of the quadrature rule employed in the discretization. As a corollary, a simple method based on singularity subtraction which has been employed by many authors is shown to be convergent.

Related papers:

title: Boundary element techniques: Theory and applications in engineering
abstract: 1 Approximate Methods.- 1.1. Introduction.- 1.2. Basic Definitions.- 1.3. Approximate Solutions.- 1.4. Method of Weighted Residuals.- 1.4.1. The Collocation Method.- 1.4.2. Method of Collocation by Subregions.- 1.5. Method of Galerkin.- 1.6. Weak Formulations.- 1.7. Inverse Problem and Boundary Solutions.- 1.8. Classification of Approximate Methods.- References.- 2 Potential Problems.- 2.1. Introduction.- 2.2. Elements of Potential Theory.- 2.3. Indirect  2.4. Direct  2.5. Boundary Element Method.- 2.6. Two-Dimensional Problems.- 2.6.1. Source Formulation.- 2.7. Poisson Equation.- 2.8.  Fundamental Solutions.- 2.12. Three-Dimensional Problems.- 2.13. Axisymmetric Problems.- 2.14. Axisymmetric Problems with Arbitrary Boundary Conditions.- 2.15. Nonlinear Materials and Boundary Conditions.- 2.15.1. Nonlinear Boundary Conditions.- References.- 3 Interpolation Functions.- 3.1. Introduction.- 3.2. Linear Elements for Two-Dimensional Problems.- 3.3. Quadratic and Higher-Order Elements.- 3.4. Boundary Elements for Three-Dimensional Problems.- 3.4.1. Quadrilateral Elements.- 3.4.2. Higher-Order Quadrilateral Elements.- 3.4.3. Lagrangian Quadrilateral Elements.- 3.4.4. Triangular Elements.- 3.4.5. Higher-Order Triangular Elements.- 3.5. Three-Dimensional Cell Elements.- 3.5.1. Tetrahedron.- 3.5.2. Cube.- 3.6. Discontinuous Boundary Elements.- 3.7.  Transforms.- 4.3. Coupled Boundary Element - Finite Difference Methods.- 4.4. Time-Dependent Fundamental Solutions.- 4.5. Two-Dimensional Problems.- 4.5.1. Constant Time Interpolation.- 4.5.2. Linear Time Interpolation.- 4.5.3. Quadratic Time Interpolation.- 4.5.4. Space Integration.- 4.6. Time-Marching Schemes.- 4.7. Three-Dimensional Problems.- 4.8. Axisymmetric Problems.- 4.9. Nonlinear Diffusion.- References.- 5 Elastostatics.- 5.1. Introduction to the Theory of Elasticity.- 5.1.1.  Fundamental Integral Statement.- 5.2.1. Somigliana Identity.- 5.3. Fundamental Solutions.- 5.4. Stresses at Internal Points.- 5.5. Boundary Integral Equation.- 5.6. Infinite and Semi-Infinite Regions.- 5.7. Numerical Implementation.- 5.8. Boundary Elements.- 5.9. System of Equations.- 5.10. Stresses and Displacements Inside the Body.- 5.11. Stresses on the Boundary.- 5.12. Surface Traction Discontinuities.- 5.13. Two-Dimensional Elasticity.- 5.14. Body Forces.- 5.14.1. Gravitational Loads.- 5.14.2. Centrifugal Load.- 5.14.3. Thermal Loading.- 5.15. Axisymmetric Problems.- 5.15.1. Extension to Nonaxisymmetric Boundary Values.- 5.16.  Boundary Integral Formulation for Inelastic Problems.- 6.1. Introduction.- 6.2. Inelastic Behavior of Materials.- 6.3. Governing Equations.- 6.4. Boundary Integral Formulation.- 6.5. Internal Stresses.- 6.6. Alternative Boundary Element Formulations.- 6.6.1.  6.6.2. Initial Stress.- 6.6.3. Fictitious Tractions and Body Forces.- 6.7. Half-Plane Formulations.- 6.8. Spatial Discretization.- 6.9. Internal Cells.- 6.10. Axisymmetric Case.- References.- 7 Elastoplasticity.- 7.1. Introduction.- 7.2. Some Simple Elastoplastic Relations.- 7.3.  Outline of Solution Techniques.- 7.5.1.  Kelvin Implementation.- 7.5.2.  Half-Plane Implementation.- 7.6. Comparison with Finite Elements.- . Rate-Dependent Constitutive Equations.- 8.3.   Time-Dependent Problems.- 8.5. No-Tension Materials.-  Integral Equations.- 9.3.1. Other Fundamental Solutions.- 9.4.  Propagation Problems.- 10.1. Introduction.- 10.2. Three-Dimensional Water Wave Propagation Problems.- 10.3.  Transient Scalar Wave Equation.- 10.7. Three-Dimensional Problems: The Retarded Potential.- 10.8. Two-Dimensional Problems.- References.- 11 Vibrations.- 11.1. Introduction.- 11.2. Governing Equations.- 11.3. Time-Dependent Integral Formulation.- 11.4. Laplace Transform  11.5. Steady-State Elastodynamics.- 11.6. Free Vibrations.- References.- 12 Further Applications in Fluid Mechanics.- 12.1.  Transient Groundwater Flow.- 12.3. Moving Interface Problems.- 12.4. Axisymmetric Bodies in Cross Flow.- 12.5. Slow Viscous Flow (Stokes Flow).- 12.6. General Viscous Flow.- 12.6.1. Steady Problems.- 12.6.2. Transient Problems.-  Coupling of Boundary Elements with Other Methods.- 13.1.  Coupling of Finite Element and Boundary Element Solutions.- 13.2.1. The Energy Approach.- 13.3. Surface Boundary Condition.- 13.4.2. Extension to Compressible Fluid.- 13.5. Approximate Boundary Elements.- 13.6. Approximate Finite Elements.- References.- 14 Computer Program for Two-Dimensional Elastostatics.- 14.1.  Subroutine MATRX.- 14.5. Subroutine FUNC.- 14.6. Subroutine SLNPD.- 14.7. Subroutine OUTPT.- 14.8. Subroutine FENC.- 14.9. Examples.- 14.9.1. Square Plate.- 14.9.2. Cylindrical Cavity Problem.- References.-  One-Dimensional Quadrature.- A.2.2. Two- and Three-Dimensional Quadrature for Rectangles and Rectangular Hexahedra.- A.2.3.

title: Boundary element methods
abstract: The boundary element method (BEM) has become a major numerical tool in scientific and engineering problem solving, with particular applications in the solution of partial differential equations in engineering. This book has been written as a self-contained reference, combining both the mathematical rigor necessary for a full understanding of BEM, and extensive examples of applications and illustrations.

title: Discrete projection methods for integral equations
abstract: Discrete projection methods are analyzed for a variety of integral equations which arise from the reformulation of boundary value problems for partial differential equations. The theory is illustrated by applying it to a variety of integral equations: Fredholm, Volterra, Cauchy singular, hypersingular and boundary integral. The material should be of interest to a large audience of graduate students, practising researchers in engineering and the sciences and applied mathematicians.

title: Boundary integral equations
abstract: This book is devoted to the basic mathematical properties of solutions to boundary integral equations and presents a systematic approach to the variational methods for the boundary integral equations arising in elasticity, fluid mechanics, and acoustic scattering theory. It may also serve as the mathematical foundation of the boundary element methods. The latter have recently become extremely popular and efficient computational tools in applications. The authors are well known for their fundamental work on boundary integral equations and related topics. This book is a major scholarly contribution to the modern theory of boundary integral equations and should be accessible and useful to a large community of mathematical analysts, applied mathematicians, engineers and scientists.

title: Linear Integral Equations
abstract: Normed Spaces.- Bounded and Compact  Riesz Theory.- Dual Systems and Fredholm Alternative.- Regularization in Dual Systems.- Potential Theory.- Singular Integral Equations.- Sobolev Spaces.- The Heat Equation.- Operator Approximations .-Degenerate Kernel Approximation.- Quadrature  Projection Methods.- Iterative Solution and Stability.- Equations of the First Kind.- Tikhonov Regularization.- Regularization by Discretization.- Inverse Boundary Value Problems.- References.- Index.

title: The Fast Solution of Boundary Integral Equations
abstract: Boundary Integral Equations.- Boundary Element Methods.- Approximation of Boundary Element Matrices.- Implementation and Numerical Examples.

title: Acoustic and Electromagnetic Equations : Integral Representations for Harmonic Problems
abstract: Preface.- Some wave equations.- Harmonic Helmholtz equation.- Integral representations and Integral equations.- Singular integral operators.- Maxwell equations and electromagnetic waves.- References.

title: Boundary Integral Methods in Fluid Mechanics
abstract: Part 1 Introduction to fluid mechanics: basic conservation laws approximate forms of the governing equations special forms of the governing equations. Part 2 Integral equation theory: classification of integral equations method of successive approximations integral equations with degenerate kernels general case of Fredholm's equation systems of integral equations. Part 3 Potential theory: basic concepts of potential theory indirect formulation regularity conditions for exterior problems. Part 4 Numerical solution of potential flow problems: boundary integral equation formulation and numerical solution of selected problems. Part 5 Boundary integral equations for low Reynolds number flow: Greens' identities hydrodynamic single- and double-layer potentials indirect formulation Lyapunov-Tauber theorem for Stokes double-layer potential dynamic properties of the singularities and their distributions. Part 6 The low Reynolds number deformation of viscous drops and gas bubbles: viscous drop deformation compound drop deformation gas bubble deformation. Part 7 Navier-Stokes equations: velocity-pressure formulation velocity-vorticity formulation.

title: Boundary Integral Equations in Elasticity Theory
abstract: Preface. Preface Z. Olesiak.  Singular Solutions and Potentials in Complex Form. 3. Complex Integral Equations of the Indirect Approach. 4. Complex Integral Equations of the Direct Approach.  Based on the Theory by Kolosov-Muskhelishvili. 5. Functions of Kolosov-Muskhelishvili and Holomorphicity Theorems. 6. Complex Variable Integral Equations. 7. Periodic Problems. 8. Doubly Periodic Problems. 9. Problems for Bonded Half-Planes and Circular Inclusion. Part III: Theory of Complex Integral Equations. 10. Complex Hypersingular and Finite-Part Integrals. 11. Complex Variable Hypersingular Equations (CVH-BIE). Part IV: Numerical Solution of Complex Variable Boundary Integral Equations. 12. Complex Variable Boundary Element Method (CV-BEM). 13. Numerical Experiments Using CV-BEM. 14. Complex Variable Method of Mechanical Quadratures (CV-MMQ). Index.

title: Precise boundary element computation of protein transport properties: Diffusion tensors, specific volume, and hydration.
abstract: A precise boundary element method for the computation of hydrodynamic properties has been applied to the study of a large suite of 41 soluble proteins ranging from 6.5 to 377 kDa in molecular mass. A hydrodynamic model consisting of a rigid protein excluded volume, obtained from crystallographic coordinates, surrounded by a uniform hydration thickness has been found to yield properties in excellent agreement with experiment. The hydration thickness was determined to be delta = 1.1 +/- 0.1 A. Using this value, standard deviations from experimental measurements are: 2% for the specific volume; 2% for the translational diffusion coefficient, and 6% for the rotational diffusion coefficient. These deviations are comparable to experimental errors in these properties. The precision of the boundary element method allows the unified description of all of these properties with a single hydration parameter, thus far not achieved with other methods. An approximate method for computing transport properties with a statistical precision of 1% or better (compared to 0.1-0.2% for the full computation) is also presented. We have also estimated the total amount of hydration water with a typical -9% deviation from experiment in the case of monomeric proteins. Both the water of hydration and the more precise translational diffusion data hint that some multimeric proteins may not have the same solution structure as that in the crystal because the deviations are systematic and larger than in the monomeric case. On the other hand, the data for monomeric proteins conclusively show that there is no difference in the protein structure going from the crystal into solution.

title: Numerical solution of boundary-integral equations for molecular electrostatics.
abstract: Numerous molecular processes, such as ion permeation through channel proteins, are governed by relatively small changes in energetics. As a result, theoretical investigations of these processes require accurate numerical methods. In the present paper, we evaluate the accuracy of two approaches to simulating boundary-integral equations for continuum models of the electrostatics of solvation. The analysis emphasizes boundary-element method simulations of the integral-equation formulation known as the apparent-surface-charge (ASC) method or polarizable-continuum model (PCM). In many numerical implementations of the ASC/PCM model, one forces the integral equation to be satisfied exactly at a set of discrete points on the boundary. We demonstrate in this paper that this approach to discretization, known as point collocation, is significantly less accurate than an alternative approach known as qualocation. Furthermore, the qualocation method offers this improvement in accuracy without increasing simulation time. Numerical examples demonstrate that electrostatic part of the solvation free energy, when calculated using the collocation and qualocation methods, can differ significantly; for a polypeptide, the answers can differ by as much as 10 kcal/mol (approximately 4% of the total electrostatic contribution to solvation). The applicability of the qualocation discretization to other integral-equation formulations is also discussed, and two equivalences between integral-equation methods are derived.

title: Predicting protein diffusion coefficients.
abstract: Diffusion coefficients for proteins in water are predicted. The numerical method developed is general enough to be applied to a wide range of protein surface shapes, from rodlike to globular. Results are presented for lysozyme and tobacco mosaic virus, and they are compared with actual data and with predictions made by less general methods.

title: Wavelet-Like Bases for the Fast Solution of Second-Kind Integral Equations
abstract: A class of vector-space bases is introduced for the sparse representation of discretizations of integral operators. An operator with a smooth, nonoscillatory kernel possessing a finite number of singularities in each row or column is represented in these bases as a sparse matrix, to high precision. A method is presented that employs these bases for the numerical solution of second-kind integral equations in time bounded by $O(n\log ^2 n)$, where n is the number of points in the discretization. Numerical results are given which demonstrate the effectiveness of the approach, and several generalizations and applications of the method are discussed.

title: Wavelet Galerkin Algorithms for Boundary Integral Equations
abstract: The implementation of a fast, wavelet-based Galerkin discretization of second kind integral equations on piecewise smooth surfaces $\Gamma\subset \IR^3$ is described. It allows meshes consisting of triangles as well as quadrilaterals. The algorithm generates a sparse, approximate stiffness matrix with $\cN=O(N(\log N)^2)$ nonvanishing entries in O(N(log N)4) operations, where N is the number of degrees of freedom on the boundary, while essentially retaining the asymptotic convergence rate of the full Galerkin scheme. A new proof of the matrix-compression estimates is given based on derivative-free kernel estimates. The condition number of the sparse stiffness matrices is bounded independently of the meshwidth. The data structure containing the compressed stiffness matrix is described in detail: it requires $O(\cN)$ memory and can be set up in $O(\cN)$ operations. Numerical experiments show that the asymptotic performance estimates apply for moderate N. Problems with N=106 degrees of freedom were computed in core on a workstation. The impact of various parameters in the compression scheme on the performance and the accuracy of the algorithm is studied.

title: THE NUMERICAL SOLUTION OF LAPLACE ’ S EQUATION IN THREE DIMENSIONS
abstract: We consider the Dirichlet problem for Laplace's equation, on a simply-connected three-dimensional region with a smooth boundary. This problem is easily converted to the solution of a Fredholm integral equation of the second kind, based on representing the harmonic solution as a double layer potential function. We solve this integral equation formulation by using Galerkin's method, with spherical harmonics as the basis functions. This approach leads to small linear systems, and once the Galerkin coefficients for the region have been calculated, the computation time is small for any particular boundary function. The major disadvantage of the method is the calculation of the Galerkin coefficients, each of which is a four-fold integral with a singular integrand. Theoretical and computational details of the method are presented.

title: A high-order algorithm for obstacle scattering in three dimensions
abstract: In this work we describe, implement and analyse in detail a high-order fully discrete spectral algorithm for solving the Helmholtz equation exterior to a bounded (sound-soft, sound-hard or absorbing) obstacle in three space dimensions, with Dirichlet, Neumann or Robin (impedance) boundary conditions. Our algorithm may be thought of as a discrete Glerkin method, but it is also equivalent to a Nystrom method after a simple transformation. We test our algorithm with extensive computational experiments on a variety of three-dimensional smooth and non-smooth obstacles with conical singularities. Our tests include the computation of scattered and far fields induced by incident plane waves. Our method is shown to be very accurate for scattering from surfaces which are globally parameterised by spherical coordinates, and tests show that it performs very much better than several ,of the well-established fast algorithms for obstacle sattering on a range of such surfaces, even some which are non-smooth. Further, we prove superalgebraic convergence of the scattered and far fields obtained using our algorithm in the case of smooth scatterers.

title: A New Spectral Boundary Integral Collocation Method for Three-Dimensional Potential Problems
abstract: In this work we propose and analyze a new global collocation method for classical second-kind boundary integral equations of potential theory on smooth simple closed surfaces $\Gamma \subset {\Bbb R}^3$. Under the assumption that $\Gamma$ is diffeomorphic to the unit sphere $\partial B$, the original equation is transferred to an equivalent one on $\partial B$ which is solved using collocation onto a nonstandard set of basis functions. The collocation points are situated on lines of constant latitude and longitude. The interpolation operator used in the collocation method is equivalent to a certain discrete orthogonal (pseudospectral) projection, and this equivalence allows us to establish the fundamental properties of the interpolation process and subsequently to prove that our collocation method is stable and super-algebraically convergent. In addition, we describe a fast method for computing the weakly singular collocation integrals and present some numerical experiments illustrating the use of the method. These show that at least for model problems the method attains an exponential rate of convergence and exhibits a good accuracy for very small numbers of degrees of freedom.

title: Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in 
${\mathbb R}^3$
abstract: In this paper we describe and analyse a class of spectral methods, based on spherical polynomial approximation, for second-kind weakly singular boundary integral equations arising from the Helmholtz equation on smooth closed 3D surfaces diffeomorphic to the sphere. Our methods are fully discrete Galerkin methods, based on the application of special quadrature rules for computing the outer and inner integrals arising in the Galerkin matrix entries. For the outer integrals we use, for example, product-Gauss rules. For the inner integrals, a variant of the classical product integration procedure is employed to remove the singularity arising in the kernel. The key to the analysis is a recent result of Sloan and Womersley on the norm of discrete orthogonal projection operators on the sphere. We prove that our methods are stable for continuous data and superalgebraically convergent for smooth data. Our theory includes as a special case a method closely related to one of those proposed by Wienert (1990) for the fast computation of direct and inverse acoustic scattering in 3D.

title: A fast, high-order algorithm for the solution of surface scattering problems: basic implementation, tests, and applications
abstract: We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates scattered fields through fast, high-order solution of the corresponding boundary integral equation. The high-order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of a novel approach which, based on high-order “two-face” equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D fast Fourier transforms (FFTs). This approach is faster and substantially more accurate, and it runs on dramatically smaller memories than other FFT and k-space methods. The present algorithm computes one matrix-vector multiplication in O(N6/5logN) to O(N4/3logN) operations, where N is the number of surface discretization points. The latter estimate applies to smooth surfaces, for which our high-order algorithm provides accurate solutions with small values of N; the former, more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling. Further, our approach exhibits super-algebraic convergence; it can be applied to smooth and nonsmooth scatterers, and it does not suffer from accuracy breakdowns of any kind. In this paper we introduce the main algorithmic components in our approach, and we demonstrate its performance with a variety of numerical results. In particular, we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds.

title: Numerical Solution of the Helmholtz Equation in 2D and 3D Using a High-Order Nyström Discretization
abstract: We show how to solve time-harmonic scattering problems by means of a high-order Nystrom discretization of the boundary integral equations of wave scattering in 2D and 3D. The novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity. Enhanced by local corrections, the new algorithm has the simplicity and speed advantages of the traditional Nystrom method, but also enjoys the advantages of high-order convergence for controlling solution error. We explain the practical details of implementing a scattering code based on a high-order Nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high-order convergence to the correct answer. We also demonstrate its performance advantages over a high-order Galerkin code.

title: Convergence and conditioning of a Nyström method for Stokes flow in exterior three-dimensional domains
abstract: Convergence and conditioning results are presented for the lowest-order member of a family of Nyström methods for arbitrary, exterior, three-dimensional Stokes flow. The flow problem is formulated in terms of a recently introduced two-parameter, weakly singular boundary integral equation of the second kind. In contrast to methods based on product integration, coordinate transformation and singularity subtraction, the family of Nyström methods considered here is based on a local polynomial correction determined by an auxiliary system of moment equations. The polynomial correction is designed to remove the weak singularity in the integral equation and provide control over the approximation error. Here we focus attention on the lowest-order method of the family, whose implementation is especially simple. We outline a convergence theorem for this method and illustrate it with various numerical examples. Our examples show that well-conditioned, accurate approximations can be obtained with reasonable meshes for a range of different geometries.

title: Quadrature methods for 2D and 3D problems
abstract: In this paper we give an overview on well-known stability and convergence results for simple quadrature methods based on low-order composite quadrature rules and applied to the numerical solution of integral equations over smooth manifolds. First, we explain the methods for the case of second-kind equations. Then we discuss what is known for the analysis of pseudodifferential equations. We explain why these simple methods are not recommended for integral equations over domains with dimension higher than one. Finally, for the solution of a two-dimensional singular integral equation, we prove a new result on a quadrature method based on product rules.

title: A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains
abstract: We present a high-order boundary integral equation solver for 3D elliptic boundary value problems on domains with smooth boundaries. We use Nystrom's method for discretization, and combine it with special quadrature rules for the singular kernels that appear in the boundary integrals. The overall asymptotic complexity of our method is O(N3/2), where N is the number of discretization points on the boundary of the domain, and corresponds to linear complexity in the number of uniformly sampled evaluation points. A kernel-independent fast summation algorithm is used to accelerate the evaluation of the discretized integral operators. We describe a high-order accurate method for evaluating the solution at arbitrary points inside the domain, including points close to the domain boundary. We demonstrate how our solver, combined with a regular-grid spectral solver, can be applied to problems with distributed sources. We present numerical results for the Stokes, Navier, and Poisson problems.

title: On Stable, Complete, and Singularity-Free Boundary Integral Formulations of Exterior Stokes Flow
abstract: A new boundary integral formulation of the second kind for exterior Stokes flow is in- troduced. The formulation is stable, complete, singularity-free, and natural for bodies of complicated shape and topology. We prove an existence and uniqueness result for the formulation for arbitrary flows and illustrate its performance via several numerical examples using a Nystrom method with Gauss-Legendre quadrature rules of different order.

title: Microhydrodynamics: Principles and Selected Applications
abstract: This text focuses on determining the motion of particles through a viscous fluid in bounded and unbounded flow. Its central theme is the mobility relation between particle motion and forces, and Lecture some pages from the more than through time reversibility means then plates arranged. A force distribution of the lamb's general information these properties stokes flow. Advances in late august and singularity, methods vanishing at infinity can be theoretical. Students can be theoretical questions and vanishing at these terms stokeslet. Application of stokes equations lorentz reciprocal theorem can. The are negligible in more general case of chemical. Then the body is to chemical engineering theory in stokes equations. Kim and pressure design methodology of catalysis thermodynamics transport phenomena on. In chemical engineering theory and graduate hours introduction to avoid indexing. Explain the stokeslet which is stokes equations.

title: Second kind integral equation formulation of Stokes' flows past a particle of arbitary shape
abstract: The problem of determining the slow viscous flow of an unbounded fluid past a single solid particle is formulated exactly as a system of linear Fredholm integral equations of the second kind for a ...

title: Singularity subtraction in the numerical solution of integral equations
abstract: The singularity subtraction technique described by Kantorovich and Krylov in [11] is designed to reduce or overcome the effect of a weakly singular kernel in the numerical solution of integral equations. First, the equation is rearranged in such a way that the singularity of the kernel is at least partially cancelled by the smoothness of the solution, and then numerical integration is applied. We present convergence results and error bounds under general conditions on the nature of the singularity and the numerical integration procedure. Numerical examples demonstrate the benefit of the singularity subtraction technique.