Inverse scattering for an exterior Dirichlet problem

The Numerical Solution of Integral Equations on the Half-Line

Numerial Methods for Volterra Integral Equations with Singular Kernels

This book provides an extensive introduction to the numerical solution of a large class of integral equations. The initial chapters provide a general framework for the numerical analysis of Fredholm integral equations of the second kind, covering degenerate kernel, projection and Nystrom methods. Additional discussions of multivariable integral equations and iteration methods update the reader on the present state of the art in this area. The final chapters focus on the numerical solution of boundary integral equation (BIE) reformulations of Laplace's equation, in both two and three dimensions. Two chapters are devoted to planar BIE problems, which include both existing methods and remaining questions. Practical problems for BIE such as the set up and solution of the discretised BIE are also discussed. Each chapter concludes with a discussion of the literature and a large bibliography serves as an extended resource for students and researchers needing more information on solving particular integral equations.

Conjugate gradient methods for the solution of boundary integral equations on a piecewise smooth boundary

We give a convergence and error analysis for a Nystrom method on a graded mesh for the numerical solution of boundary integral equations for the harmonic Dirichlet problem in plane domains with corners.

A domain decomposition method (DDM) for the solution of boundary integral equations for impenetrable objects is presented. The approach uses a transmission condition which is based on the near-range couplings. In order to obtain close approximations of the global solution on the subdomains, the comparatively strong near-zone interactions are taken into account to avoid unphysical reflections from the domain interfaces. The algorithm can be accelerated by fast integral solvers such as the multilevel fast multipole method (MLFMM) by creating local subtrees, which coincide with the different computing domains. This concept is embedded in an inner-outer iterative solution algorithm in which the DDM acts as a preconditioner in the inner iterations. By introducing small overlap regions, the convergence of the iterative solver is further improved. This results in a very robust DDM which can be integrated into existing codes with reasonable effort and which is suited for parallelization. Numerical results for various open and closed objects demonstrate the effectiveness and the excellent performance of the proposed DDM for the solution of electromagnetic radiation and scattering problems.

A fast multigrid solution of boundary integral equations

A discrete collocation method for Symm's integral equation on curves with corners

We consider the Dirichlet problem for Laplace's equation on planar domains with corners.A Nyström method is used to solve the corresponding double layer boundary integral equation.The Nyström method is based on the trapezoidal rule with a graded mesh method as in [13], but we adopt a generalized mesh grading.An improved convergence is observed.

A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson’s equation on irregular exterior regions. The method requires $O(N\log N)$operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.

The major problem encountered by the solution of boundary integral equations is related to the nonuniqueness of its solution. The problem is also known as internal resonance problem. As a remedy the combined-field integral equation (CFIE) technique is widely used which contains a linear combination of the magnetic and electric field integral equation to provide a unique stable solution. A second effective technique is the constrained conjugate gradient method (CCG) that minimizes a cost functional consisting of two terms. The first term is the error norm with respect to boundary integral equation, while the second term is the error norm with respect to the interior equation over a closed interior surface. This paper presents the efficiency of both approaches. The results are compared for the solution of a sphere.

The motion of shallow water, unlike deep water, is influenced by the bottom topology. In order to obtain the velocity potential at any given point of the surface at any given time, we apply boundary integral techniques and solve a system of two integral equations, one expresses the velocity potential at the free surface and on the bottom, the other expresses the streamfunction at the free surface and on the bottom. Boundary integrals can produce very accurate results for shallow water equations but they are too complex to solve analytically and must be solved numerically. In this paper, we use three different numerical methods to compute the solutions to these integral equations: by Gaussian elimination which has high compuational cost; by iteration to construct the Neumann series which converges; and by a preconditioned version of the iteration which effectively lowers the total computation cost.

We study the numerical solution of the boundary integral equation formulation for the scattering of time-harmonic electromagnetic waves from infinite cylinders. For smooth boundaries of the cylinder cross section we describe a Nystro¨m, a collocation and a Galerkin method based on an approximation by trigonometric polynomials on an equidistant mesh. For smooth data in each of the three methods the convergence is exponential. From the three approaches the Nystro¨m method is the most efficient since it requires the least computational effort. For cross sections with corners we develop a Nystro¨m method on a graded mesh based on the idea of transforming the nonsmooth case to a smooth periodic case via an appropriate substitution. We conclude the paper with some considerations on the corresponding three dimensional problem.

A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions

We present a new formulation of the integral equation (IE) method for models with variable background conductivity. This method overcomes the standard limitation of the conventional IE method related to the use of a horizontally layered background only. The new IE method still employs the Green's functions for a horizontally layered 1‐D model. However, the new method allows us to use an inhomogeneous background with the IE method. The method was carefully tested for modeling the electromagnetic (EM) field for complex structures with known variable background conductivity. It can find wide application in modeling EM data for multiple geological models with some common geoelectrical features. As an example, we consider marine controlled source EM (MCSEM) application for simulating EM response of a petroleum reservoir in the presence of a salt dome structure.