Numerical Solution of a Surface Hypersingular Integral Equation by Piecewise Linear Approximation and Collocation Methods

Abstract

A linear hypersingular integral equation is considered on a surface (closed or nonclosed with a boundary). This equation arises when the Neumann boundary value problem for the Laplace equation is solved by applying the method of boundary integral equations and the solution is represented in the form of a double-layer potential. For such an equation, a numerical scheme is constructed by triangulating the surface, approximating the solution by a piecewise linear function, and applying the collocation method at the vertices of the triangles approximating the surface. As a result, a system of linear equations is obtained that has coefficients expressed in terms of integrals over partition cells containing products of basis functions and a kernel with a strong singularity. Analytical formulas for finding these coefficients are derived. This requires the computation of the indicated integrals. For each integral, a neighborhood of the singular point is traversed so that the system of linear equations approximates the integrals of the unknown function at the collocation points in the sense of the Hadamard finite part. The method is tested on some examples.