Finite element approximation of Maxwell eigenproblems on curved Lipschitz polyhedral domains (Q1030667)

The authors study finite element approximations of Maxwell eigenvalue problems on a non-convex Lipschitz domain with curved boundary. The asymptotic analysis is established based on the lowest-order edge element discretization of the Maxwell eigenvalue problem with smooth and strictly positive coefficients. The authors study the following issues concerning the independent error estimates for eigenvalues and eigenfunctions: {\parindent=5mm \begin{itemize}\item[1.] The kernel of the curl-operator is the range of the gradient operator. Thus the eigenspace of the curl-operator belonging to the zero eigenvalue is infinite diemensional. \item[2.] Since the computational domains are not included in the domain where the continuous model is defined, the finite element method turns to be a nonconforming approximation. The authors use the zero-extensions of continuous and discrete eigenfunctions to the exterior of their domains, and prove that the consistency terms due to the domain discrepancy converge to zero. \item[3.] Using the technique of zero-extensions, the authors prove new Helmholtz-type decompositions and the discrete compactness of discrete divergence-free functions, which prove fundamental instruments for a priori error estimates of the eigenvalues and eigenfunctions. \end{itemize}} The authors prove optimal error estimates for the eigenvalues and eigenfunctions by estimating the discrepancy between the computational domains and the original domain. This paper is a good reference for a numerical treatments of Maxwell's equations and finite element approximations on domains with curved boundaries.