Numerical solution of boundary integral equations on curvilinear polygons

The Numerical Solution of Integral Equations on the Half-Line

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

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.

Numerial Methods for Volterra Integral Equations with Singular Kernels

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

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.

Numerical solution of boundary integral equations on non smooth boundaries

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.

In this paper we consider scattering due to a metallic cylinder which is in the field of a wire carrying a periodic current. The aim of the paper is to obtain information such as the location and shape of the cylinder with a knowledge of a far-field measurement in between the wire and the cylinder. The same analysis is applicable in acoustics in the situation that the cylinder is a soft-wall body and the wire is a line source. The associated direct problem in this situation is an exterior Dirichlet problem for the Helmholtz equation in two dimensions. We present an improved low-frequency estimate for the solution of this problem using integral equation methods, and our calculations on inverse scattering are accurate to this estimate. The far-field measurements are related to the solutions of boundary integral equations in the low-frequency situation. These solutions can be expressed in terms of mapping function which maps the exterior of the unknown curve onto the exterior of a unit disk. The coefficients of the Laurent expansion of the conformal transformations can be related to the far-field coefficients. The first far-field coefficient leads to the calculation of the distance between the source and the cylinder. The other coefficients are determined by placing the source in a different location and using the corresponding new far-field measurements.

Fuzzy aspects of the boundary element method

This paper deals with approximate solutions to integral equations arising in boundary value problems for the biharmonic equation in simply connected piecewise smooth domains. The approximation method considered demonstrates excellent convergence even in the case of boundary conditions discontinuous at corner points. In an application we obtain very accurate approximations for some characteristics of two‐dimensional Stokes flow in non‐smooth domains. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An efficient algorithm for the numerical evaluation of boundary integral equations

A fast multigrid solution of boundary integral equations

Non-equivalence of the conventional boundary integral formulation and its elimination for plane elasticity problems