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

2018/10/02 by Deyue Zhang, Fenglin Sun, Zhang, Deyue +5
Engineering · Mathematics · #Composite Material Mechanics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1810.01055

openalex publication_date 2018/10/02 · openalex created_date 2018/10/12 · openalex updated_date 2026/08/01

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.

Related

• A spectral approach to a constrained optimization problem for the Helmholtz equation in unbounded domains
• Regularized integral formulation of mixed Dirichlet-Neumann problems
• A fast Fourier transform based direct solver for the Helmholtz problem
• Isogeometric solution of Helmholtz equation with Dirichlet boundary condition: numerical experiences
• Inverse boundary value problem for the Helmholtz equation: Multi-level approach and iterative reconstruction