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- Beilina, Larisa, 1970, et al.
(författare)
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Convergence of explicit p1 Finite-Element Solutions to Maxwell’s Equations
- 2020
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Ingår i: Springer Proceedings in Mathematics and Statistics. - Cham : Springer International Publishing. - 2194-1017 .- 2194-1009. ; 328, s. 91-103
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Konferensbidrag (refereegranskat)abstract
- This paper is devoted to the numerical validation of an explicit finite-difference scheme for the integration in time of Maxwell’s equations in terms of the sole electric field. The space discretization is performed by the standard P1 finite element method assorted with the treatment of the time-derivative term by a technique of the mass-lumping type. The rigorous reliability analysis of this numerical model was the subject of authors’ another paper [2]. More specifically such a study applies to the particular case where the electric permittivity has a constant value outside a sub-domain, whose closure does not intersect the boundary of the domain where the problem is defined. Our numerical experiments in two-dimension space certify that the convergence results previously derived for this approach are optimal, as long as the underlying CFL condition is satisfied.
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- Beilina, Larisa, 1970, et al.
(författare)
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Explicit P 1 Finite Element Solution of the Maxwell-Wave Equation Coupling Problem with Absorbing b. c.
- 2024
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Ingår i: Mathematics. - 2227-7390. ; 12:7
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Tidskriftsartikel (refereegranskat)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.
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- Beilina, Larisa, 1970, et al.
(författare)
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On the Maxwell-wave equation coupling problem and its explicit finite-element solution
- 2023
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Ingår i: Applications of Mathematics. - 1572-9109 .- 0862-7940. ; 68:1, s. 75-98
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Tidskriftsartikel (refereegranskat)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.
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- Beilina, Larisa, 1970, et al.
(författare)
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On the Maxwell-wave equation coupling problem and its explicit finite-element solution
- 2022
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Ingår i: Applications of Mathematics. - : Institute of Mathematics, Czech Academy of Sciences. - 0862-7940. ; 68:1, s. 75-98
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Tidskriftsartikel (refereegranskat)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.
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