Gauss-compatible Galerkin schemes for time-dependent Maxwell equations

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 numerical solutions to the time-dependent Maxwell system. In particular, we formulate a compatibility condition relative to the preservation of genuinely oscillating modes that takes the form of a generalized commuting diagram, and we show that compatible schemes satisfy convergence estimates leading to long-time stability with respect to stationary solutions. These findings are applied by specifying 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 conserving, unlike existing DG discretizations where the introduction of a dissipative penalty term is needed to avoid the presence of spurious modes.