GAUSS-COMPATIBLE GALERKIN SCHEMES FOR TIME-DEPENDENT MAXWELL EQUATIONS

Abstract. In this article we propose a unified analysis for conforming and non-conforming finite element methods that provides a partial answer to the problem of preserving discrete divergence constraints when computing numer-ical solutions to the time-dependent Maxwell system. In particular, we formu-late a compatibility condition relative to the preservation of genuinely oscil-lating modes that takes the form of a generalized commuting diagram, and we show that compatible schemes satisfy convergence estimates leading to long-time stability near stationary solutions. We next apply these findings by spec-ifying compatible formulations for several classes of Galerkin methods, such as the usual curl-conforming finite elements and the centered discontinuous Galerkin (DG) scheme. We also propose a new conforming/non-conforming Galerkin (Conga) method where fully discontinuous solutions are computed by embedding the general structure of curl-conforming finite elements into larger DG spaces. In addition to naturally preserving one of the Gauss laws in a strong sense, the Conga method is both spectrally correct and energy conserv-ing, unlike existing DG discretizations where the introduction of a dissipative penalty term is necessary to avoid the presence of spurious modes. 1