High order corrected trapezoidal rules for a class of singular integrals

TL;DR

This paper introduces high order corrected trapezoidal quadrature rules tailored for singular integrals with point singularities, enhancing accuracy through Taylor series expansions and correction weights.

Contribution

It develops a novel family of quadratures that achieve high order accuracy for singular integrals by correcting trapezoidal rule weights based on series expansion.

Findings

Achieves high order accuracy for singular integrals.

Effective correction method for integrals with point singularities.

Applicable to boundary integral formulations of Laplace problems.

Abstract

We present a family of high order trapezoidal rule-based quadratures for a class of singular integrals, where the integrand has a point singularity. The singular part of the integrand is expanded in a Taylor series involving terms of increasing smoothness. The quadratures are based on the trapezoidal rule, with the quadrature weights for Cartesian nodes close to the singularity judiciously corrected based on the expansion. High order accuracy can be achieved by utilizing a sufficient number of correction nodes around the singularity to approximate the terms in the series expansion. The derived quadratures are applied to the Implicit Boundary Integral formulation of surface integrals involving the Laplace layer kernels.