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Upwind discontinuous Galerkin space discretization and locally implicit time integration for linear Maxwell's equations


Authors: Marlis Hochbruck and Andreas Sturm
Journal: Math. Comp.
MSC (2010): Primary 65M12, 65M15; Secondary 65M60, 65J10
DOI: https://doi.org/10.1090/mcom/3365
Published electronically: June 19, 2018
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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 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.


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Additional Information

Marlis Hochbruck
Affiliation: Institute for Applied and Numerical Analysis, Karlsruhe Institute of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany
Email: marlis.hochbruck@kit.edu

Andreas Sturm
Affiliation: Institute for Applied and Numerical Analysis, Karlsruhe Institute of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany
Email: andreas.sturm@kit.edu

DOI: https://doi.org/10.1090/mcom/3365
Keywords: Locally implicit methods, time integration, upwind fluxes discontinuous Galerkin finite elements, error analysis, energy techniques, Maxwell's equations
Received by editor(s): May 26, 2017
Received by editor(s) in revised form: December 29, 2017, and January 25, 2018
Published electronically: June 19, 2018
Additional Notes: We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.
Article copyright: © Copyright 2018 American Mathematical Society

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