Semi-analytical calculation of the singular and hypersingular integrals for discrete Helmholtz operators in 2D BEM

TL;DR

This paper derives semi-analytical formulas for singular and hypersingular integrals in 2D Helmholtz boundary element methods, facilitating more accurate and efficient computation of boundary operators in electromagnetic, acoustic, and quantum applications.

Contribution

It provides quasi-closed-form expressions for singular and hypersingular integrals in 2D Helmholtz BEM, improving computational implementation.

Findings

Formulas enable precise evaluation of boundary integrals.

Reduction to quasi-closed-form expressions simplifies numerical implementation.

Potential for enhanced accuracy in electromagnetic, acoustic, and quantum simulations.

Abstract

Approximate solutions to elliptic partial differential equations with known kernel can be obtained via the boundary element method (BEM) by discretizing the corresponding boundary integral operators and solving the resulting linear system of algebraic equations. Due to the presence of singular and hypersingular integrals, the evaluation of the operator matrix entries requires the use of regularization techniques. In this work, the singular and hypersingular integrals associated with first-order Galerkin discrete boundary operators for the two-dimensional Helmholtz equation are reduced to quasi-closed-form expressions. The obtained formulas may prove useful for the implementation of the BEM in two-dimensional electromagnetic, acoustic and quantum mechanical problems.