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 u on connected regions D in R with boundary S The boundary S is allowed to be smooth or piecewise smooth and we let f K j K Ng 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 uN Superconvergence results for uN are given for S a smooth surface and for a special type of re nement strategy for the triangulation We show u uN is O log at the collocation node points with the mesh size for f Kg 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 but the numerical experiments suggest the result is O for the error at the collocationpoints especially for S a smooth surface The numerical integration of the collocation integrals is discussed and extended numerical examples are given for problems involving both smooth and piecewise smooth surfaces