Jaap van der Vegt & Dietmar Gallistl
The research objective is to analyze discontinuous Galerkin methods for the Maxwell eigenproblem with very weak smoothness requirements. Discontinuous Galerkin (DG) finite element methods are well suited to solve the Maxwell eigenvalue problem numerically, for instance due their suitability for complex geometries, parallel computing and hp-adaptation, which consists of local mesh refinement and the local adjustment of the polynomial order. For a relative smooth solution, the a priori error analysis of discontinuous Galerkin dicretizations for the time-harmonic Maxwell equations is well developed. However, the strong smoothness assumptions in these a priori error estimates often are not realistic, since at sharp corners and material interfaces the solution of the Maxwell eigenproblem contains singularities and is non-smooth. These problems require therefore a detailed a priori error analysis with minimal smoothness requirements, which is a challenging research area. In a later stage of this project, we also will consider accurate and efficient a posteriori error estimators for non-smooth Maxwell eigenvalue problems. This will greatly facilitate the development and use of hp-adaptive discontinuous Galerkin discretizations for the Maxwell eigenproblem.