Geodesic
On the Maxwell-wave equation coupling problem and its explicit finite-element solution
Applications of Mathematics, Tome 68 (2023) no. 1, pp. 75-98
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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.
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.
DOI : 10.21136/AM.2022.0210-21
Classification : 65M12, 65M22, 65M60
Keywords: constant magnetic permeability; dielectric permittivity; explicit scheme; finite element; mass lumping; Maxwell-wave equation 
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Beilina, Larisa; Ruas, Vitoriano. On the Maxwell-wave equation coupling problem and its explicit finite-element solution. Applications of Mathematics, Tome 68 (2023) no. 1, pp. 75-98. doi: 10.21136/AM.2022.0210-21

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