In this paper we prove the discrete compactness property for a discontinuous Galerkin approximation of Maxwell's system on quite general tetrahedral meshes. As a consequence, a discrete Friedrichs inequality is obtained and the convergence of the discrete eigenvalues to the continuous ones is deduced using the theory of collectively compact operators. Some numerical experiments confirm the theoretical predictions.
Mots clés : DG method, Maxwell's system, discrete compactness, eigenvalue approximation
@article{M2AN_2006__40_2_413_0,
author = {Creus\'e, Emmanuel and Nicaise, Serge},
title = {Discrete compactness for a discontinuous {Galerkin} approximation of {Maxwell's} system},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {413--430},
publisher = {EDP-Sciences},
volume = {40},
number = {2},
year = {2006},
doi = {10.1051/m2an:2006017},
zbl = {1112.78020},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2006017/}
}
TY - JOUR AU - Creusé, Emmanuel AU - Nicaise, Serge TI - Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 413 EP - 430 VL - 40 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2006017/ DO - 10.1051/m2an:2006017 LA - en ID - M2AN_2006__40_2_413_0 ER -
%0 Journal Article %A Creusé, Emmanuel %A Nicaise, Serge %T Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 413-430 %V 40 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2006017/ %R 10.1051/m2an:2006017 %G en %F M2AN_2006__40_2_413_0
Creusé, Emmanuel; Nicaise, Serge. Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 2, pp. 413-430. doi : 10.1051/m2an:2006017. http://www.numdam.org/articles/10.1051/m2an:2006017/
[1] , Collectively compact operator approximation theory. Prentice Hall (1971). | MR
[2] and, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Method. Appl. Sci. 21 (1998) 519-549. | Zbl
[3] and, Eigenvalue solvers for electromagnetic fields in cavities, in High performance scientific and engineering computing, H.-J. Bungartz, F. Durst, and C. Zenger, Eds., Lect. Notes Comput. Sc., Springer, Berlin 8(1999). | MR
[4] ,, and, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749-1779. | Zbl
[5] , and, Characterization of the singular part of the solution of Maxwell's equations in a polyhedral domain. RAIRO Modél. Math. Anal. Numér. 32 (1998) 485-499. | Zbl
[6] and, -theory of the Maxwell operator in arbitrary domains. Russ. Math. Surv. 42 (1987) 75-96. | Zbl
[7] , Fortin operator and discrete compactness for edge elements. Numer. Math. 87 (2000) 229-246. | Zbl
[8] , and, Discrete compactness for p and hp 2d edge finite elements. Math. Mod. Meth. Appl. S. 13 (2003) 1673-1687. | Zbl
[9] ,, and, Discrete compactness for the hp version of rectangular edge finite elements. ICES Report 04-29, University of Texas, Austin (2004). | Zbl
[10] and, Mixed and hybrid finite element methods. Springer, New York (1991). | MR | Zbl
[11] , Spectral approximation of linear operators. Academic Press, New York (1983). | MR | Zbl
[12] , The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | MR | Zbl
[13] and, Singularities of electromagnetic fields in polyhedral domains. Arch. Rational Mech. Anal. 151 (2000) 221-276. | Zbl
[14] , Benchmark computations for Maxwell equations for the approximation of highly singular solutions. Technical report, University of Rennes 1. http://perso.univ-rennes1.fr/monique.dauge/core/index.html
[15] ,, and, Maxwell eigenvalues and discrete compactness in two dimensions. Comput. Math. Appl. 40 (2000) 589-605. | Zbl
[16] and, On the solution of time-harmonic scattering problems for Maxwell's equations. SIAM J. Math. Anal. 27 (1996) 1597-1630. | Zbl
[17] and, High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem. Philos. T. Roy. Soc. A 362 (2004) 493-524. | Zbl
[18] , Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237-339. | Zbl
[19] ,, and, Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math. 100 (2005) 485-518. | Zbl
[20] , and, Mixed discontinuous Galerkin approximation of the Maxwell operator: Non-stabilized formulation. J. Sci. Comput. 22 (2005) 315-346. | Zbl
[21] , On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. U. Tokyo IA 36 (1989) 479-490. | Zbl
[22] and, On the validity of Friedrichs' inequalities. Math. Scand. 54 (1984) 17-26. | Zbl
[23] , Initial boundary value problems in Mathematical Physics. John Wiley, New York (1988). | Zbl
[24] and, A discontinuous Galerkin method on refined meshes for the 2d time-harmonic Maxwell equations in composite materials. Preprint Macs, University of Valenciennes, 2004. J. Comput. Appl. Math. (to appear). | MR | Zbl
[25] , Finite element methods for Maxwell's equations. Numer. Math. Scientific Comp., Oxford Univ. Press, New York (2003). | Zbl
[26] and, Discrete compactness and the approximation of Maxwell’s equations in . Math. Comp. 70 (2000) 507-523. | Zbl
[27] , Spectral approximation for compact operators. Math. Comp. 29 (1975) 712-725. | Zbl
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