ANALYTICAL BEHAVIOR OF SOLUTIONS OF BOUNDARY INTEGRAL-EQUATIONS FOR A NONSMOOTH REGION

The Dirichlet problem for Helmholtz's equation in a domain Ω exterior to some bounded smooth boundary g{cyrillic} in two dimensions may be solved by means of a combined potential of the single and double layers. In this paper, the problem arising from allowing corner points on the boundary is investigated. The resulting noncompact operator is effectively split into singular and compact parts. By using the Mellin transforms, the equation can be converted into some Cauchy-type singular integral equations. Consequently, the singular form of the solution is found in terms of rβ at a corner with 0>β>1. As a first step toward developing new numerical methods for the problem, one typical example is presented to demonstrate the slow convergence of existing methods without any modifications. Then the mesh-grading technique designed for singular equations is successfully implemented to restore the order of convergence. © 1990 Oxford University Press.