Non-conforming Crouzeix-Raviar element approximation for Stekloff eigenvalues in inverse scattering
This work provides a theoretical foundation and numerical validation for using non-conforming finite elements on a specific non-selfadjoint eigenvalue problem, which is incremental for the field of numerical analysis.
The authors extend Strang's lemma to handle non-conforming Crouzeix-Raviart element discretization of a non-selfadjoint Stekloff eigenvalue problem from inverse scattering, proving convergence and error estimates. Numerical examples on uniform and adaptive meshes demonstrate efficiency for computing both real and complex eigenvalues.
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.