SUPERCONVERGENT APPROXIMATIONS TO THE SOLUTION OF A BOUNDARY INTEGRAL-EQUATION ON POLYGONAL DOMAINS

The solution to the interior Dirichlet problem for Laplace's equation in a two-dimensional domain is the double layer potential of a double layer distribution satisfying a second kind boundary integral equation. This may be solved numerically by Galerkin's method using piecewise polynomials of degree r. The iterated Galerkin solution may then be calculated, which gives an order 2r plus 2 approximation if the domain is smooth. However if the domain is polygonal, the corners cause singularities which degrade the order of convergence. Here it is shown that a suitable grading of the mesh near the corners almost restores the rate of convergence when the error is measured in the unifor norm. We m then describe a second means of calculating a higher order approximation from the Galerkin solution. This is cheaper to calculate than the iterated Galerkin solution, but maintains the same order of convergence.