Structure preserving schemes for Maxwell and PIC: conservation of charge and long-time stability

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Résumé : In this talk 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 identify a set of conditions that a Maxwell solver must satisfy to preserve a compatible discrete Gauss law and in the case of a pure Maxwell problem we give a characterization of a compatible source approximation operator that takes the form of a generalized commuting diagram leading to long-time error and stability estimates.

We next apply these findings by specifying charge conserving schemes for several classes of Galerkin methods such as (i) the usual curl-conforming finite elements, (ii) the centered discontinuous Galerkin (DG) scheme and (iii) a new conforming/non-conforming Galerkin method that shares several advantages with both conforming and DG schemes.

These results are also extended in the case where the Maxwell solver is coupled with a PIC scheme, and numerical results in 2d demonstrate some of the advantages of the proposed methods.