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Fully Discrete A-ø Finite Element Method for Maxwell’s Equations with a Nonlinear Boundary Condition

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Equations of mathematical physics and other areas of application Partial differential equations, boundary value problems

Published online by Cambridge University Press:  10 November 2015

Tong Kang ,
Ran Wang ,
Tao Chen  and
Huai Zhang
Tong Kang
Affiliation:
Department of Applied Mathematics, School of Sciences, Communication University of China, Beijing, 100024, China.
Ran Wang
Affiliation:
Department of Applied Mathematics, School of Sciences, Communication University of China, Beijing, 100024, China.
Tao Chen
Affiliation:
Department of Applied Mathematics, School of Sciences, Communication University of China, Beijing, 100024, China.
Huai Zhang*
Affiliation:
Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing, 100049, China.
*
*Corresponding author. Email address: [email protected] (H. Zhang)
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Abstract

In this paper we present a fully discrete A-ø 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 x n = n x (|E x n|α-1E x n) with α ∈ (0,1]. We prove the existence and uniqueness of the solutions of the proposed A-ø scheme and derive the error estimates. Finally, we present some numerical experiments to verify the theoretical result.

Keywords

Maxwell’s equationsnonlinear Silver-Müller boundary conditionA-ø methodfull discretizationnodal elementserror estimates

MSC classification

Secondary: 35Q60: PDEs in connection with optics and electromagnetic theory 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 4 , November 2015 , pp. 605 - 633
Copyright
Copyright © Global-Science Press 2015 

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