A Fourier-Bessel method with a regularization strategy for the boundary value problems of the Helmholtz equation

TL;DR

This paper introduces a Fourier-Bessel method with regularization for solving boundary value problems of the Helmholtz equation, providing stability, convergence analysis, and numerical validation in smooth domains.

Contribution

It develops a new regularized Fourier-Bessel approach with stability and convergence guarantees for Helmholtz boundary problems.

Findings

Established a lower bound for the operator's smallest singular value

Proved stability and convergence of the regularized solution

Numerical experiments confirm the method's effectiveness

Abstract

This paper is concerned with the Fourier-Bessel method for the boundary value problems of the Helmholtz equation in a smooth simply connected domain. Based on the denseness of Fourier-Bessel functions, the problem can be approximated by determining the unknown coefficients in the linear combination. By the boundary conditions, an operator equation can be obtained. We derive a lower bound for the smallest singular value of the operator, and obtain a stability and convergence result for the regularized solution with a suitable choice of the regularization parameter. Numerical experiments are also presented to show the effectiveness of the proposed method.