A Finite Element Method for a Curlcurl-Graddiv Eigenvalue Interface Problem

Abstract

In this paper we propose and study a finite element method for a curlcurl-graddiv eigenvalue interface problem. Its solution may be of piecewise non-$H^1$. We would like to approximate such a solution in an $H^1$-conforming finite element space. With the discretizations of both curl and div operators of the underlying eigenvalue problem in two finite element spaces, the proposed method is essentially a standard $H^1$-conforming element method, up to element bubbles which can be statically eliminated at element levels. We first analyze the proposed method for the related source interface problem by establishing the stability and the error bounds. We then analyze the underlying eigenvalue interface problem, and we obtain the error bounds $\mathcal{O}(h^{2r_0})$ for eigenvalues which correspond to eigenfunctions in $\prod_{j=1}^J (H^r(\Omega_j))^3\hookrightarrow (H^{r_0}(\Omega))^3$ space, where the piecewise regularity $r$ and the global regularity $r_0$ may belong to the most interesting interval $[0,1]$.