Homogenization of the Maxwell Equations: Case II. Nonlinear conductivity

Niklas Wellander

Applications of Mathematics (2002)

  • Volume: 47, Issue: 3, page 255-283
  • ISSN: 0862-7940

Abstract

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The Maxwell equations with uniformly monotone nonlinear electric conductivity in a heterogeneous medium, which may be non-periodic, are homogenized by two-scale convergence. We introduce a new set of function spaces appropriate for the nonlinear Maxwell system. New compactness results, of two-scale type, are proved for these function spaces. We prove existence of a unique solution for the heterogeneous system as well as for the homogenized system. We also prove that the solutions of the heterogeneous system converge weakly to the solution of the homogenized system. Furthermore, we prove corrector results, important for numerical implementations.

How to cite

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Wellander, Niklas. "Homogenization of the Maxwell Equations: Case II. Nonlinear conductivity." Applications of Mathematics 47.3 (2002): 255-283. <http://eudml.org/doc/32526>.

@article{Wellander2002,
abstract = {The Maxwell equations with uniformly monotone nonlinear electric conductivity in a heterogeneous medium, which may be non-periodic, are homogenized by two-scale convergence. We introduce a new set of function spaces appropriate for the nonlinear Maxwell system. New compactness results, of two-scale type, are proved for these function spaces. We prove existence of a unique solution for the heterogeneous system as well as for the homogenized system. We also prove that the solutions of the heterogeneous system converge weakly to the solution of the homogenized system. Furthermore, we prove corrector results, important for numerical implementations.},
author = {Wellander, Niklas},
journal = {Applications of Mathematics},
keywords = {nonlinear PDEs; Maxwell’s equations; nonlinear conductivity; homogenization; existence of solution; unique solution; two-scale convergence; corrector results; heterogeneous materials; compactness result; non-periodic medium; nonlinear PDEs; Maxwell equations; nonlinear conductivity; homogenization; existence of solution; unique solution; two-scale convergence; corrector results; heterogeneous materials; compactness result; non-periodic medium},
language = {eng},
number = {3},
pages = {255-283},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Homogenization of the Maxwell Equations: Case II. Nonlinear conductivity},
url = {http://eudml.org/doc/32526},
volume = {47},
year = {2002},
}

TY - JOUR
AU - Wellander, Niklas
TI - Homogenization of the Maxwell Equations: Case II. Nonlinear conductivity
JO - Applications of Mathematics
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 3
SP - 255
EP - 283
AB - The Maxwell equations with uniformly monotone nonlinear electric conductivity in a heterogeneous medium, which may be non-periodic, are homogenized by two-scale convergence. We introduce a new set of function spaces appropriate for the nonlinear Maxwell system. New compactness results, of two-scale type, are proved for these function spaces. We prove existence of a unique solution for the heterogeneous system as well as for the homogenized system. We also prove that the solutions of the heterogeneous system converge weakly to the solution of the homogenized system. Furthermore, we prove corrector results, important for numerical implementations.
LA - eng
KW - nonlinear PDEs; Maxwell’s equations; nonlinear conductivity; homogenization; existence of solution; unique solution; two-scale convergence; corrector results; heterogeneous materials; compactness result; non-periodic medium; nonlinear PDEs; Maxwell equations; nonlinear conductivity; homogenization; existence of solution; unique solution; two-scale convergence; corrector results; heterogeneous materials; compactness result; non-periodic medium
UR - http://eudml.org/doc/32526
ER -

References

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