A-priori and a-posteriori error estimates for discontinuous Galerkin method of the Maxwell eigenvalue problem

This paper is devoted to a-priori and a-posteriori error analysis of discontinuous Galerkin (DG) method for the Maxwell eigenvalue problem. The discrete compactness of DG space is proved so that the Babuška and Osborn spectral approximation theory can be applicable in the a-priori error analysis. Then we prove the optimal error estimates for DG eigenfunctions in mesh-dependent norm and DG eigenvalues. A special contribution of this work is to prove that the error in $L^2$-norm for smooth eigenfunctions is of higher order than that in mesh-dependent norm, so that the DG eigenvalues can approximate the true eigenvalues from upper. Another contribution of this work is to provide a-posteriori error analysis for the DG method. A reliable a-posteriori error estimator is analyzed. The upper bound property of DG eigenvalues and the robustness of adaptive methods are verified through numerical experiments.