A Penalty-Free or Penalty-Factor-Free DG Method with a Locally Reconstructed Curl Operator for the Maxwell Eigenproblem

Abstract

A new discontinuous Galerkin (DG) method is proposed and analyzed for the Maxwell eigenproblem, featuring a local reconstruction of the curl operator in a discontinuous finite element space. The proposed method can be penalty-free or penalty-factor-free, depending on which discontinuous finite element space the curl operator is locally reconstructed in. The new DG method is recast into a saddle-point problem so that it can be analyzed from the Babuška-Osborn theory for the finite element approximation of the spectrum of the compact operator, and the convergence and the optimal error estimates are then obtained; the discrete eigenmodes are spurious-free and spectral-correct. We provide numerical results to illustrate the proposed method.