Convergence analysis of two finite element methods for the modified Maxwell’s Steklov eigenvalue problem

Bo Gong

doi:10.1051/m2an/2022001

Abstract

The modified Maxwell’s Steklov eigenvalue problem is a new problem arising from the study of inverse electromagnetic scattering problems. In this paper, we investigate two finite element methods for this problem and perform the convergence analysis. Moreover, the monotonic convergence of the discrete eigenvalues computed by one of the methods is analyzed.

Highlights

The Steklov eigenvalue problem is governed by the elliptic equation with the eigen-parameter in the boundary condition
It has many applications in physics, e.g., surface waves [4] and stability of mechanical oscillators immersed in a viscous fluid [13]
A new application was considered in [10] using the Steklov eigenvalues as a target signature in nondestructive testing

Summary

Introduction

The Steklov eigenvalue problem is governed by the elliptic equation with the eigen-parameter in the boundary condition. While in [15], a specific finite element method was considered and a convergence order of the corresponding discrete eigenvalues were obtained. Sh does not, i.e., it has a range not surface-divergence-free Despite of this difference between S︀h and Sh, they display similar numerical behaviour (see [15]). The major difficulty in the analysis lies in the fact that the range of Sh is not surfacedivergence-free To this end, we define a solution operator slightly different than that from [11] so that its domain and range are the L2 space other than the surface-divergence-free space. We define a solution operator slightly different than that from [11] so that its domain and range are the L2 space other than the surface-divergence-free space Both finite element methods (with S︀h and Sh) under this L2-to-L2 framework are analyzed.

Preliminary

Finite element method using Sh

Additional approximation properties

Conclusion