Upwind discontinuous Galerkin space discretization and locally implicit time integration for linear Maxwell’s equations. (English) Zbl 1462.65144
Authors’ abstract: This paper is dedicated to the full discretization of linear Maxwell’s equations, where the space discretization is carried out with a discontinuous Galerkin (dG) method on a locally refined spatial grid. For such problems, explicit time integrators are inefficient due to their strict CFL condition stemming from the fine grid elements. In the last few years, this issue of so-called grid-induced stiffness was successfully tackled with locally implicit time integrators. So far, these methods are limited to unstabilized (central fluxes) dG methods. However, stabilized (upwind fluxes) dG schemes provide many benefits and thus are a popular choice in applications. In this paper, we construct a new variant of a locally implicit time integrator using an upwind fluxes dG discretization on the coarse part of the grid. The construction is based on a rigorous error analysis which shows that the stabilization operators have to be split differently than the Maxwell operator. Moreover, our earlier analysis of a central fluxes locally implicit method based on semigroup theory [M. Hochbruck and A. Sturm, SIAM J. Numer. Anal. 54, No. 5, 3167–3191 (2016; Zbl 1457.65111)] applies but does not yield optimal convergence rates. In this paper, we rigorously prove the stability and provide error bounds of the new method with optimal rates in space and time by means of an energy technique for a suitably defined modified error.
Reviewer: Dimitra Antonopoulou (Chester)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65J10 | Numerical solutions to equations with linear operators |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 78A25 | Electromagnetic theory (general) |
| 35Q61 | Maxwell equations |
Keywords:
locally implicit methods; time integration; upwind fluxes discontinuous Galerkin finite elements; error analysis; energy techniques; Maxwell’s equationsCitations:
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