Journal Article

A Novel Spectral Approximation and Error Estimation for Transmission Eigenvalues in Spherical Domains

Tan, Ting, An, Jing, Zhang, Zhimin

2023

Overview

In this paper, we propose and analyze an efficient spectral-Galerkin method based on a mixed formulation with dimension reduction for the Helmholtz transmission eigenvalue problem in spherical domains. By introducing an auxiliary function, we rewrite the original problem as an equivalent fourth-order coupled form in spherical coordinates. Using the properties of spherical harmonic and Laplace–Beltrami operator, we further decompose the original problem into a series of decoupled one-dimensional fourth-order linear eigenvalue problems, for which a new mixed variational formulation and its discretization is developed. For error estimates of numerical eigenvalues and eigenfunctions, we recall the spectral theory of compact operators. Towards this end, we derive the essential polar conditions, define a class of weighted Sobolev spaces, and most importantly, prove a sequence of two compact embedding properties for the weighted Sobolev spaces, based on which the spectral theory of compact operators for the variational formulation and discrete system can be established. Finally, some numerical examples are presented to confirm the theoretical error analysis and the efficiency of our algorithm.

• Publisher: Springer US,Springer Nature B.V
• Subject: Algorithms / Approximation / Computational Mathematics and Numerical Analysis / Coordinate transformations / Discrete systems / Efficiency / Eigenvalues / Eigenvectors / Error analysis / Galerkin method / Linear operators / Mathematical analysis / Mathematical and Computational Engineering / Mathematical and Computational Physics / Mathematics / Mathematics and Statistics / Operators (mathematics) / Sobolev space / Spectral theory / Spherical coordinates / Spherical harmonics / Theoretical