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Fully discrete finite element method for Maxwell's equations with nonlinear conductivity

Stephane Durand (UGent) and Marian Slodicka (UGent)
(2011) IMA JOURNAL OF NUMERICAL ANALYSIS. 31(4). p.1713-1733
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01D28807
Abstract
In this paper we present a numerical scheme to solve nonlinear Maxwell’s equations based on backward Euler discretization in time and curl-conforming finite elements in space. The nonlinearity is due to a field-dependent conductivity in the form of a power law. The system under study is hyperbolic and due to the nonlinear conductivity it lacks strong estimates of the second time derivative. We are able to prove convergence of our numerical scheme based on boundedness of the second derivative in the dual space. Convergence of the nonlinear term is based on the Minty–Browder technique. We also present the error estimate for the fully discretized problem and support the theory by some numerical experiments.
Keywords
POLYHEDRAL DOMAINS, SUPERCONDUCTORS, TIME-DISCRETIZATION SCHEME, CONVERGENCE, MODEL, Maxwell's equations, backward Euler, convergence, edge elements, error estimates

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Chicago
Durand, Stephane, and Marian Slodicka. 2011. “Fully Discrete Finite Element Method for Maxwell’s Equations with Nonlinear Conductivity.” Ima Journal of Numerical Analysis 31 (4): 1713–1733.
APA
Durand, S., & Slodicka, M. (2011). Fully discrete finite element method for Maxwell’s equations with nonlinear conductivity. IMA JOURNAL OF NUMERICAL ANALYSIS, 31(4), 1713–1733.
Vancouver
1.
Durand S, Slodicka M. Fully discrete finite element method for Maxwell’s equations with nonlinear conductivity. IMA JOURNAL OF NUMERICAL ANALYSIS. 2011;31(4):1713–33.
MLA
Durand, Stephane, and Marian Slodicka. “Fully Discrete Finite Element Method for Maxwell’s Equations with Nonlinear Conductivity.” IMA JOURNAL OF NUMERICAL ANALYSIS 31.4 (2011): 1713–1733. Print.
@article{1218404,
  abstract     = {In this paper we present a numerical scheme to solve nonlinear Maxwell{\textquoteright}s equations based on backward Euler discretization in time and curl-conforming finite elements in space. The nonlinearity is due to a field-dependent conductivity in the form of a power law. The system under study is hyperbolic and due to the nonlinear conductivity it lacks strong estimates of the second time derivative. We are able to prove convergence of our numerical scheme based on boundedness of the second derivative in the dual space. Convergence of the nonlinear term is based on the Minty--Browder technique. We also present the error estimate for the fully discretized  problem and support the theory by some numerical experiments.},
  author       = {Durand, Stephane and Slodicka, Marian},
  issn         = {0272-4979},
  journal      = {IMA JOURNAL OF NUMERICAL ANALYSIS},
  keyword      = {POLYHEDRAL DOMAINS,SUPERCONDUCTORS,TIME-DISCRETIZATION SCHEME,CONVERGENCE,MODEL,Maxwell's equations,backward Euler,convergence,edge elements,error estimates},
  language     = {eng},
  number       = {4},
  pages        = {1713--1733},
  title        = {Fully discrete finite element method for Maxwell's equations with nonlinear conductivity},
  url          = {http://dx.doi.org/10.1093/imanum/drr007},
  volume       = {31},
  year         = {2011},
}

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