Gauss-compatible Galerkin schemes for time-dependent Maxwell equations
Résumé
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/nonconforming
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.
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