A Fourier-Bessel method with a regularization strategy for the boundary value problems of the Helmholtz equation
It offers a theoretically grounded numerical method for Helmholtz boundary value problems, but the contribution is incremental as it extends existing Fourier-Bessel techniques with regularization.
The paper develops a Fourier-Bessel method with a regularization strategy for solving boundary value problems of the Helmholtz equation in smooth domains, providing a stability and convergence result and demonstrating effectiveness through numerical experiments.
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.