On the Maxwell-wave equation coupling problem and its explicit finite-element solution

Larisa Beilina; Vitoriano Ruas

Applications of Mathematics (2023)

  • Volume: 68, Issue: 1, page 75-98
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Beilina, Larisa, and Ruas, Vitoriano. "On the Maxwell-wave equation coupling problem and its explicit finite-element solution." Applications of Mathematics 68.1 (2023): 75-98. <http://eudml.org/doc/299401>.

@article{Beilina2023,
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.},
author = {Beilina, Larisa, Ruas, Vitoriano},
journal = {Applications of Mathematics},
keywords = {constant magnetic permeability; dielectric permittivity; explicit scheme; finite element; mass lumping; Maxwell-wave equation},
language = {eng},
number = {1},
pages = {75-98},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the Maxwell-wave equation coupling problem and its explicit finite-element solution},
url = {http://eudml.org/doc/299401},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Beilina, Larisa
AU - Ruas, Vitoriano
TI - On the Maxwell-wave equation coupling problem and its explicit finite-element solution
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 1
SP - 75
EP - 98
AB - 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.
LA - eng
KW - constant magnetic permeability; dielectric permittivity; explicit scheme; finite element; mass lumping; Maxwell-wave equation
UR - http://eudml.org/doc/299401
ER -

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