Immersed Finite Element Method for Eigenvalue Problem

TL;DR

This paper develops and analyzes an immersed finite element method for elliptic eigenvalue problems with interfaces, proving stability, convergence, and optimal eigenvalue approximation, supported by numerical experiments.

Contribution

It introduces a stable and convergent immersed finite element method for eigenvalue problems with interfaces, using Crouzeix-Raviart nonconforming elements, with proven optimal convergence rates.

Findings

Spectral analysis applies to the immersed interface model.

The method achieves optimal eigenvalue convergence.

Numerical experiments confirm theoretical results.

Abstract

We consider the approximation of elliptic eigenvalue problem with an immersed interface. The main aim of this paper is to prove the stability and convergence of an immersed finite element method (IFEM) for eigenvalues using Crouzeix-Raviart $P_1$-nonconforming approximation. We show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem. We analyze the IFEM for elliptic eigenvalue problem with an immersed interface and derive the optimal convergence of eigenvalues. Numerical experiments demonstrate our theoretical results.