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research article

High order geometric methods with splines: fast solution with explicit time-stepping for Maxwell equations

Kapidani, Bernard  
•
Vazquez Hernandez, Rafael  
February 9, 2023
Journal of Computational Physics

We introduce a high-order spline geometric approach for the initial boundary value problem for Maxwell's equations. The method is geometric in the sense that it discretizes in structure preserving fashion the two de Rham sequences of differential forms involved in the formulation of the continuous system. Both the Ampere--Maxwell and the Faraday equations are required to hold strongly, while to make the system solvable two discrete Hodge star operators are used. By exploiting the properties of the chosen spline spaces and concepts from exterior calculus, a non-standard explicit in time formulation is introduced, based on the solution of linear systems with matrices presenting Kronecker product structure, rather than mass matrices as in the standard literature. These matrices arise from the application of the exterior (wedge) product in the discrete setting, and they present Kronecker product structure independently of the geometry of the domain or the material parameters. The resulting scheme preserves the desirable energy conservation properties of the known approaches. The computational advantages of the newly proposed scheme are studied both through a complexity analysis and through numerical experiments in three dimensions.

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Type
research article
DOI
10.1016/j.jcp.2023.112440
Author(s)
Kapidani, Bernard  
Vazquez Hernandez, Rafael  
Date Issued

2023-02-09

Published in
Journal of Computational Physics
Volume

493

Article Number

112440

Subjects

maxwell equations

•

differential forms

•

kronecker product

•

geometric method

•

splines

•

isogeometric analysis

•

conforming b-splines

•

element exterior calculus

•

isogeometric analysis

•

tetrahedral grids

•

schemes

Note

Preprint

URL

Version éditeur

https://doi.org/10.1016/j.jcp.2023.112440
Editorial or Peer reviewed

REVIEWED

Written at

EPFL

EPFL units
MNS  
FunderGrant Number

FNS

HOGAEMS n.200021_188589

Available on Infoscience
February 23, 2023
Use this identifier to reference this record
https://infoscience.epfl.ch/handle/20.500.14299/195051
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