PIECEWISE POLYNOMIAL COLLOCATION FOR BOUNDARY INTEGRAL-EQUATIONS

This paper considers the numerical solution of boundary integral equations of the second kind for Laplace's equation Delta u = 0 on connected regions D in R(3) with boundary S. The boundary S is allowed to be smooth or piecewise smooth, and we let {Delta K \ 1 less than or equal to K less than or equal to N} be a triangulation of S. The numerical method is collocation with approximations which are piecewise quadratic in the parametrization variables, leading to a numerical solution u(N). Superconvergence results for u(N) are given for S a smooth surface and for a special type of refinement strategy for the triangulation. We show that u - u(N) is O (delta(4) log delta) at the collocation node points, with delta being the mesh size for {Delta(K)}. Error analyses are given are given for other quantities, and an important error analysis is given for the approximation of S by piecewise quadratic interpolation on each triangular element, with S either smooth or piecewise smooth. The convergence result we prove is only O (delta(2)) but the numerical experiments suggest the result is O (delta(4)) for the error at the collocation points, especially when S is a smooth surface. The numerical integration pf the collocation integrals is discussed, and extended numerical examples are given for problems involving both smooth and piecewise smooth surfaces.