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Preconditioned implicit time integration schemes for Maxwell’s equations on locally refined grids

Hochbruck, Marlis 1; Köhler, Jonas 1; Kumbhar, Pratik M. 1
1 Institut für Angewandte und Numerische Mathematik (IANM), Karlsruher Institut für Technologie (KIT)

Abstract:

In this paper, we consider an efficient implementation of higher-order implicit time integration schemes for spatially discretized linear Maxwell’s equations on locally refined meshes.
In particular, our interest is in problems where only a few of the mesh elements are small while the majority of the elements is much larger. We suggest to approximate the solution of the linear systems arising in each time step by a preconditioned Krylov subspace method, e.g., the quasi-minimal residual method by Freund and Nachtigal [13].
Motivated by the analysis of locally implicit methods by Hochbruck and Sturm [25], we show how to construct a preconditioner in such a way that the number of iterations required by the Krylov subspace method to achieve a certain accuracy is bounded independently of the diameter of the small mesh elements. We prove this behavior by using Faber polynomials and complex approximation theory.
The cost to apply the preconditioner consists of the solution of a small linear system, whose dimension corresponds to the degrees of freedom within the fine part of the mesh (and its next coarse neighbors). If this dimension is small compared to the size of the full mesh, the preconditioner is very efficient.
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Volltext §
DOI: 10.5445/IR/1000148078
Veröffentlicht am 18.07.2022
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsmonat/-jahr 06.2022
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000148078
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 17 S.
Serie CRC 1173 Preprint ; 2022/29
Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019)
Externe Relationen Siehe auch
Schlagwörter Maxwell’s equations, higher-order time integration, locally refined mesh, Krylov subspace methods, preconditioning, error analysis
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