Nystrom Discretization of Integral Equations on Planar Curves with Corners

摘要

The Nystrom method can produce ill-conditioned systems of linear equations and inaccurate results when applied to integral equations on domains with corners. This defect can already be seen in the simple case of the integral equations arising from the Neumann problem for Laplace's equation. We explain the origin of this instability and show that a straightforward modification to the Nystrom scheme, which renders it mathematically equivalent to Galerkin discretization, corrects the difficulty without incurring the computational penalty associated with Galerkin methods. We also present the results of numerical experiments showing that highly accurate solutions of integral equations on domains with corners can be obtained, irrespective of whether their solutions exhibit bounded or unbounded singularities, assuming that proper discretizations are used.