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Fully discrete finite element scheme for Maxwell’s equations with non-linear boundary condition. (English) Zbl 1211.78030

A study is done of a full Maxwell’s system accompanied with a non-linear degenerate boundary condition, which represents a generalization of the classical Silver-Müller condition for a non-perfect conductor. The relationship between the normal components of electric \(E\) and magnetic \(H\) field obeys the following power law \(\nu \times H=\nu \times (|E\times \nu |^{\alpha - 1}E\times \nu )\) for some \(\alpha \in (0,1]\). The existence and uniqueness of a weak solution are established in a suitable function spaces under the minimal regularity assumptions on the boundary \(\Gamma \) and the initial data \(E_{0}\) and \(H_{0}\). A non-linear time discrete approximation scheme is designed and convergence of the approximations to a weak solution is proved. The error estimates are derived for the time discretization. As a next step the fully discrete problem is studied using curl-conforming edge elements and the corresponding error estimates are derived. Finally some numerical experiments are presented.

MSC:

78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
78A30 Electro- and magnetostatics
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs

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GetDP; Gmsh
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References:

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