Computing weakly singular and near-singular integrals over curved boundary elements

TL;DR

This paper introduces advanced algorithms for accurately computing weakly singular and near-singular integrals in 3D Helmholtz boundary element methods, improving numerical solutions for scattering problems.

Contribution

It develops novel algorithms combining Newton's method, singularity subtraction, continuation, and transplanted Gauss quadrature for curved boundary elements.

Findings

Algorithms achieve high accuracy for quadratic basis functions.

Numerical experiments validate effectiveness in scattering by two half-spheres.

Method improves computational reliability for near-singular integrals.

Abstract

We present algorithms for computing weakly singular and near-singular integrals arising when solving the 3D Helmholtz equation with curved boundary elements. These are based on the computation of the preimage of the singularity in the reference element's space using Newton's method, singularity subtraction, the continuation approach, and transplanted Gauss quadrature. We demonstrate the accuracy of our method for quadratic basis functions and quadratic triangles with several numerical experiments, including the scattering by two half-spheres.