A Nystr�m method for boundary integral equations in domains 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 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.

Two sparse preconditioned iterative methods are presented to solve dense linear systems arising in the solution of two-dimensional boundary integral equations. In the first method, the sparse preconditioner is constructed simply by choosing a small block of elements in the coefficient matrix of a dense linear system. The two-grid method falls into this category when the dense linear system arises from the Nyström method for a second kind boundary integral equation. In the second method, the sparse preconditioner is obtained through condensation of the coefficient matrix by discrete Fourier transforms, which can be implemented efficiently using fast Fourier transforms. Both iterative methods involve only $O(N^2 )$ arithmetic operations per iteration and converge rapidly when the dense linear systems arise from quadrature methods for boundary integral equations arising in two-dimensional problems. The author’s numerical experiments demonstrate the computational efficiency of each method.

A fast multigrid solution of boundary integral equations

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.

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.

A quantative comparison between the boundary integral equation (BIE) method and the finite difference (FD) method is presented in which each technique is applied to an elliptic boundary‐value problem (BVP) containing a boundary singularity. Two types of singularity have previously been analysed theoretically, namely those due to a discontinuous boundary potential, which we shall refer to as S1, and those due to a sudden change from the specification of boundary potential flux, an S2 singularity. In this paper the analysis is presented for a third type of boundary singularity, namely an S3 singularity: that arising from a discontinuous boundary flux. Such a condition is frequently encountered in the field of heat transfer where, for example, a system or pipe has a change of lagging material.In general, it is found that the BIE method is superior, with regards to the computational time required to achieve a certain degree of accuracy, over standard FD methods even when there is a boundary singularity. Further, the BIE method determines the solution near the singularity much more accurately than the FD method. The FD method does, however, have advantages for a very restrictive class of problems; for example, when the boundary conditions are of the Dirichlet type and the boundary geometry is rectangular. In this case an optimum relaxation parameter can easily be obtained. A soon as Neumann conditions are prescribed, the BIE is far more efficient than the FD, whatever the boundary geometry.It is concluded that, for fast, accurate solutions of general Laplacian boundary‐value problems, the BIE is appreciably superior to the FD and this is even more pronounced when there is a boundary singularity.

One popular approach to solve the sea surface scattering and propagation in a ducting environment is the parabolic wave equation (PWE) method. An alternative method is the boundary integral equations (BIE) method. The implementation of the BIE in inhomogeneous media (ducting environments) is not straightforward, however, since the Green's function for such a medium is not usually known. In this paper, a closed-form approximation of the Green's function for a two-dimensional (2-D) ducting environment made up of a duct having a linear-square refractive index profile below a medium of constant refractive index, recently published, is used. This paper demonstrates how the BIE method can model the combined effects of surface roughness and medium inhomogeneity. Furthermore, it illustrates its capability of accurately predicting scattering in all directions including backscattering. Then, the PWE combined with the Split-Step Fourier (SSF) method is compared with this method.

Acoustic wave scattering by scatterers in layered media is studied using a new method that combines the hybrid ray–mode and boundary integral equation methods. The latter is used to formulate the scattering process and the former is used to provide the Green’s function of the layered environment. Following standard procedures of the boundary integral equation method, integral equations are formulated first for field distributions along the boundaries of the obstacles. By expressing these unknown distributions in terms of appropriate basis functions, the integral equations are then reduced to algebraic equations that are solved numerically. In the integral equations, it is necessary to compute the Green’s function of the layered environment for various arrangements of locations of source and receiver. None of these conventional approaches (rays, modes, and spectral integration) to evaluate the Green’s function are satisfactory for all possible arrangements. Only the newly developed hybrid ray–mode method is best suited for this purpose because it combines rays and modes self-consistently within a single framework and optimizes the advantages of each. Numerical implementation illustrates these aspects.

The superconvergence of a Galerkin collocation method for first kind boundary integral equations on smooth curves

The principles and development of the boundary integral equation (BIE) method for solving engineering problems are reviewed, with particular emphasis on applications in three-dimensional stress analysis. Its use in problems of this type is illustrated with the aid of three examples, one a classical stress concentration problem, the other two involving cracked components. Compared with a finite-element method, the BIE approach is shown to be capable of being both more accurate and more economical to use in terms of competing resources required and the cost of data preparation. These advantages may, however, be less clear-cut when predictions of stresses and displacements at a large number of points within a component are required, or when the component concerned has a relatively high surface-to-volume ratio.

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

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.

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.