Two-Dimensional Quadrature for Functions with a Point Singularity on a Triangular Region

Abstract

We consider the numerical integration of functions with point singularities over a planar wedge S using isoparametric piecewise polynomial interpolation of the function and the wedge. Such integrals often occur in solving boundary integral equations using the collocation method.To obtain the same order of convergence as is true with uniform meshes for smooth functions, we introduce an adaptive refinement of the triangulation of S. Error analyses and several examples are given for a certain type of adaptive refinement.