The Rectangle Rule for Computing Hilbert Integral In Boundary Element Methods

Abstract

The generalized composite rectangle (constant) rule for the computation of Cauchy principal value integral with the singular kernel cot[(x−s)/2] is discussed. Our study is based on the investigation of the point-wise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate of the midpoint rule is higher than what is globally possible. We show that the superconvergence rate of the composite midpoint rule occurs at certain local coordinate of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis.