Applications of Mathematics, Vol. 68, No. 1, pp. 75-98, 2023
On the Maxwell-wave equation coupling problem and its explicit finite-element solution
Larisa Beilina, Vitoriano Ruas
Received September 29, 2021. Published online June 22, 2022. OPEN ACCESS
Abstract: It is well known that in the case of constant dielectric permittivity and magnetic permeability, the electric field solving the Maxwell's equations is also a solution to the wave equation. The converse is also true under certain conditions. Here we study an intermediate situation in which the magnetic permeability is constant and a region with variable dielectric permittivity is surrounded by a region with a constant one, in which the unknown field satisfies the wave equation. In this case, such a field will be the solution of Maxwell's equation in the whole domain, as long as proper conditions are prescribed on its boundary. We show that an explicit finite-element scheme can be used to solve the resulting Maxwell-wave equation coupling problem in an inexpensive and reliable way. Optimal convergence in natural norms under reasonable assumptions holds for such a scheme, which is certified by numerical exemplification.
Keywords: constant magnetic permeability; dielectric permittivity; explicit scheme; finite element; mass lumping; Maxwell-wave equation
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Affiliations: Larisa Beilina (corresponding author), Department of Mathematical Sciences, Chalmers University of Technology and Gothenburg University, SE-42196 Gothenburg, Sweden, e-mail: larisa@chalmers.se; Vitoriano Ruas, Institut Jean Le Rond d'Alembert, UMR 7190 CNRS - Sorbonne Université, F-75005, Paris, France, e-mail: vitoriano.ruas@upmc.fr