Non-conforming Crouzeix-Raviar element approximation for Stekloff eigenvalues in inverse scattering

TL;DR

This paper applies the non-conforming Crouzeix-Raviart element method to solve a Stekloff eigenvalue problem in inverse scattering, providing convergence analysis and numerical validation on uniform and adaptive meshes.

Contribution

It extends Strang lemma to non-selfadjoint, non-H^{1}-elliptic problems and demonstrates the method's effectiveness for complex eigenvalues.

Findings

Convergence of discrete eigenvalues and eigenfunctions is established.

Numerical examples confirm efficiency on uniform and adaptive meshes.

Method accurately computes real and complex eigenvalues.

Abstract

In this paper, we use the non-conforming Crouzeix-Raviart element method to solve a Stekloff eigenvalue problem arising in inverse scattering. The weak formulation corresponding to this problem is non-selfadjoint and does not satisfy $H^{1}$-elliptic condition,and its Crouzeix-Raviart element discretization does not meet the Strang lemma condition. We use the standard duality techniques to prove an extension of Strang lemma. And we prove the convergence and error estimate of discrete eigenvalues and eigenfunctions using the spectral perturbation theory for compact operators. Finally, we present some numerical examples not only on uniform meshes but also in an adaptive refined meshes to show that the Crouzeix-Raviart method is efficient for computing real and complex eigenvalues as expected.