Volume 8, Issue 4
Fully Discrete $A$-$ϕ$ Finite Element Method for Maxwell's Equations with a Nonlinear Boundary Condition

Numer. Math. Theor. Meth. Appl., 8 (2015), pp. 605-633.

Published online: 2015-08

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• Abstract

In this paper we present a fully discrete $A$-$ϕ$ finite element method to solve Maxwell's equations with a nonlinear 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 the electric field $E$ and the magnetic field $H$ obeys a power-law nonlinearity of the type $H × n = n × (|E × n|^{α-1}E × n)$ with $α ∈ (0,1]$. We prove the existence and uniqueness of the solutions of the proposed $A$-$ϕ$ scheme and derive the error estimates. Finally, we present some numerical experiments to verify the theoretical result.

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@Article{NMTMA-8-605, author = {}, title = {Fully Discrete $A$-$ϕ$ Finite Element Method for Maxwell's Equations with a Nonlinear Boundary Condition}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2015}, volume = {8}, number = {4}, pages = {605--633}, abstract = {

In this paper we present a fully discrete $A$-$ϕ$ finite element method to solve Maxwell's equations with a nonlinear 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 the electric field $E$ and the magnetic field $H$ obeys a power-law nonlinearity of the type $H × n = n × (|E × n|^{α-1}E × n)$ with $α ∈ (0,1]$. We prove the existence and uniqueness of the solutions of the proposed $A$-$ϕ$ scheme and derive the error estimates. Finally, we present some numerical experiments to verify the theoretical result.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2015.m1413}, url = {http://global-sci.org/intro/article_detail/nmtma/12425.html} }
TY - JOUR T1 - Fully Discrete $A$-$ϕ$ Finite Element Method for Maxwell's Equations with a Nonlinear Boundary Condition JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 605 EP - 633 PY - 2015 DA - 2015/08 SN - 8 DO - http://doi.org/10.4208/nmtma.2015.m1413 UR - https://global-sci.org/intro/article_detail/nmtma/12425.html KW - AB -

In this paper we present a fully discrete $A$-$ϕ$ finite element method to solve Maxwell's equations with a nonlinear 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 the electric field $E$ and the magnetic field $H$ obeys a power-law nonlinearity of the type $H × n = n × (|E × n|^{α-1}E × n)$ with $α ∈ (0,1]$. We prove the existence and uniqueness of the solutions of the proposed $A$-$ϕ$ scheme and derive the error estimates. Finally, we present some numerical experiments to verify the theoretical result.

Tong Kang, Ran Wang, Tao Chen & Huai Zhang. (2020). Fully Discrete $A$-$ϕ$ Finite Element Method for Maxwell's Equations with a Nonlinear Boundary Condition. Numerical Mathematics: Theory, Methods and Applications. 8 (4). 605-633. doi:10.4208/nmtma.2015.m1413
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