.lOUHNAI. OF COMPGTATIONAL PHYSICS 42, 77-98 (1981) The Numerical Solution of Plane Potential Problems by Improved Boundary Integral Equation Methods D. B. INCHAM,* P. J. HEGGS,’ AND M. MANZOOR* Departments of *Applied Mathematical Studies, and ‘Chemical Engineering. Unicersity of Leeds, Leeds LS2 9J7, England Received September 9, 19SO A modified boundary integral (BIE) method which facilitates accurate solution of Lapiacian boundary-value problems is presented. This method is designed specifically for treatment of problems in which singularities occur on the interface between two regions with different physical properties, and is illustrated by application to two physical probiems. Analytic expressions for the integrals arising in the piecewise-linear and piecewise-quadratic BIE approximations are also presented. These analytical expressions afford an appreciable reduction in computational time when compared with previously employed quadrature for mulae. Elliptic boundary-value problems arising from the examination of physical situations encountered in engineering and mathematical physics are, in general. intractable by analytical treatment. Although various numerical techniques have been proposed for the solution of such problems, e.g., the finite difference 1i I. finite element [ 21, and boundary integral equation (BIE) [3 ] methods, standard forms of these techniques tend to yield inaccurate solutions for problems involving boundary singularities. Consequently, the possibility of modifying these numerica! techniques to give special treatment to singular points, and thereby to obtain solutions which converge more rapidly has received considerable attention i 3-i 01. Symm [37 devised a modification of the BIE method which can successfully treat boundary singularities in two-dimensional Laplacian problems. The results obtained by employing this method offer considerable improvement over those given by Gaierkin methods modified by either mesh refinement near the singularity. 01 inclusion of terms having the analytical form of the singularity [3,6]. The Fresent investigation considers problems in which the boundary singuiarities occur on the interface between two regions with different physical properties, e.g., different thermal conductivities in heat diffusion problems ( 111, and different dielectric permittivities in electromagnetics [4]. Blue 1141 discussed how the BIE techniques may be applied to multiple-region problems and indicated that this would require a significant change in data structure in comparison with single-region problems. In this study these BIB 1-l 002 ; 999 I :‘H 1,C307c077- 22so2.00: ri Copyright :C !98 I by Acsdemlc Pr~sn, lrc Ali rights of rcproducmn I” any :bm rcsrved.