On the convergence of collocation methods for boundary integral equations on polygons

Abstract

The integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions. Convergence proofs and error estimates for these approximation methods have been only available in the following cases: Fredholm integral equations of the second kind [4], [7], one-dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves [2], [3], [17], [26]—[29], and some special results on the classical Neumann integral equation of potential theory for polygonal plane domains [5], [8], [9]. Here we give convergence proofs for collocation with piecewise linear trial functions for Neumann’s integral equation and Symm’s integral equation on plane curves with corners. We derive asymptotic error estimates in Sobolev norms and analyze the effect of graded meshes.