Explicit P 1 Finite Element Solution of the Maxwell-Wave Equation Coupling Problem with Absorbing b. c

Larisa Beilina () and Vitoriano Ruas
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Larisa Beilina: Department of Mathematical Sciences, Chalmers University of Technology, 41296 Gothenburg, Sweden
Vitoriano Ruas: Institut Jean Le Rond d’Alembert, UMR 7190 CNRS-Sorbonne Université, F-75005 Paris, France

Mathematics, 2024, vol. 12, issue 7, 1-34

Abstract: In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell’s equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell’s equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell’s equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space.

Keywords: absorbing boundary conditions; coupling problem; wave equation; Maxwell’s equations; dielectric permittivity; explicit scheme; finite elements; lumped mass; piecewise linear (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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