Non-conforming finite element methods for transmission eigenvalue problem

TL;DR

This paper develops non-conforming finite element methods for the Helmholtz transmission eigenvalue problem, providing error estimates and numerical validation for the approach.

Contribution

It introduces a weak formulation for the problem and analyzes non-conforming finite element approximations with proven error estimates.

Findings

Error estimates for discrete eigenvalues using specific non-conforming elements

Validation of the approach through numerical examples

Effective solution method for transmission eigenvalue problems

Abstract

The transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory. In this paper, for the Helmholtz transmission eigenvalue problem, we give a weak formulation which is a nonselfadjoint linear eigenvalue problem. Based on the weak formulation, we first discuss the non-conforming finite element approximation, and prove the error estimates of the discrete eigenvalues obtained by the Adini element, Morley-Zienkiewicz element, modified-Zienkiewicz element et. al. And we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem.