Piecewise Polynomial Collocation for Boundary Integral Equations

Abstract

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 ${\bf R}^3 $ with boundary S. The boundary S is allowed to be smooth or piecewise smooth, and we let $\{ \Delta _K |1 \leq K \leq 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 of the collocation integrals is discussed, and extended numerical examples are given for problems involving both smooth and piecewise smooth surfaces.