Analysis of immersed finite element methods for eigenvalue problems arising from heterogeneous media

Abstract

Eigenvalue analysis plays an important role in various fields. Computing eigensolutions is essential to interpret the dynamic interaction between the structures. Also, eigenvalue analysis is applied to stability analysis for many physical problems such as thermoelastic problems and fluid-solid interaction problems. In this dissertation, we analyze immersed finite element methods for eigenvalue problems arising from heterogeneous media. The first part 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 problems with an interface and derive the optimal convergence of eigenvalues. Numerical experiments demonstrate our theoretical results. The second part is the approximation of eigenvalue problems for elasticity equations with interface. This kind of problems can be efficiently discretized by using IFEM. By adding jump terms across the edges, the discretization yields a stable and locking free scheme. The stability and the optimal convergence of the IFEM for elasticity problems with interface are proved by adapting spectral analysis methods for the classical eigenvalue problem. Numerical experiments demonstrate our theoretical results.