Two-Grid Iteration Methods for Linear Integral Equations of the Second Kind on Piecewise Smooth Surfaces in $\mathbb{R}^3 $

Abstract

The numerical solution of integral equations of the second kind on surfaces in $\mathbb{R}^3 $ often leads to large linear systems that must be solved by iteration. An especially important class of such equations is boundary integral equation (BIE) reformulations of elliptic partial differential equations; and, in this paper BIEs of the second kind are considered for Laplace’s equation. The numerical methods used are based on piecewise polynomial isoparametric interpolation over the surface and the surface is also approximated by such interpolation. Two-grid iteration methods are considered for (1) integral equations with a smooth kernel function, (2) BIEs over smooth surfaces, and (3) BIEs over piecewise smooth surfaces. In the last case, standard two-grid iteration does not perform well, and a modified two-grid iteration method is proposed and examined empirically.