We present and analyze an interior penalty method for the numerical discretization of the indefinite time-harmonic Maxwell equations in mixed form. The method is based on the mixed discretization of the curl-curl operator developed in [Houston et al., J. Sci. Comp. 22 (2005) 325-356] and can be understood as a non-stabilized variant of the approach proposed in [Perugia et al., Comput. Methods Appl. Mech. Engrg. 191 (2002) 4675-4697]. We show the well-posedness of this approach and derive optimal a priori error estimates in the energy-norm as well as the -norm. The theoretical results are confirmed in a series of numerical experiments.
Mots-clés : discontinuous Galerkin methods, mixed methods, time-harmonic Maxwell's equations
@article{M2AN_2005__39_4_727_0,
author = {Houston, Paul and Perugia, Ilaria and Schneebeli, Anna and Sch\"otzau, Dominik},
title = {Mixed discontinuous {Galerkin} approximation of the {Maxwell} operator : the indefinite case},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {727--753},
publisher = {EDP-Sciences},
volume = {39},
number = {4},
year = {2005},
doi = {10.1051/m2an:2005032},
mrnumber = {2165677},
zbl = {1087.65106},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an:2005032/}
}
TY - JOUR AU - Houston, Paul AU - Perugia, Ilaria AU - Schneebeli, Anna AU - Schötzau, Dominik TI - Mixed discontinuous Galerkin approximation of the Maxwell operator : the indefinite case JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2005 SP - 727 EP - 753 VL - 39 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2005032/ DO - 10.1051/m2an:2005032 LA - en ID - M2AN_2005__39_4_727_0 ER -
%0 Journal Article %A Houston, Paul %A Perugia, Ilaria %A Schneebeli, Anna %A Schötzau, Dominik %T Mixed discontinuous Galerkin approximation of the Maxwell operator : the indefinite case %J ESAIM: Modélisation mathématique et analyse numérique %D 2005 %P 727-753 %V 39 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2005032/ %R 10.1051/m2an:2005032 %G en %F M2AN_2005__39_4_727_0
Houston, Paul; Perugia, Ilaria; Schneebeli, Anna; Schötzau, Dominik. Mixed discontinuous Galerkin approximation of the Maxwell operator : the indefinite case. ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 4, pp. 727-753. doi : 10.1051/m2an:2005032. http://archive.numdam.org/articles/10.1051/m2an:2005032/
[1] and, Hierarchic -edge element families for Maxwell’s equations on hybrid quadrilateral/triangular meshes. Comput. Methods Appl. Mech. Engrg. 190 (2001) 6709-6733. | Zbl
[2] ,, and, Vector potentials in three-dimensional non-smooth domains. Math. Models Appl. Sci. 21 (1998) 823-864. | Zbl
[3] ,, and, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749-1779. | Zbl
[4] and, Edge finite elements for the approximation of Maxwell resolvent operator. ESAIM: M2AN 36 (2002) 293-305. | Numdam | Zbl
[5] and, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15, Springer-Verlag, New York (1994). | MR | Zbl
[6] , and, Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal. 37 (2000) 1542-1570. | Zbl
[7] , The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | Zbl
[8] and, Modeling of electromagnetic absorption/scattering problems using -adaptive finite elements. Comput. Methods Appl. Mech. Engrg. 152 (1998) 103-124. | Zbl
[9] and, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7 (1997) 957-991. | Zbl
[10] , Finite elements in computational electromagnetism. Acta Numerica 11 (2002) 237-339. | Zbl
[11] , and, -DGFEM for Maxwell’s equations, in Numerical Mathematics and Advanced Applications ENUMATH 2001, F. Brezzi, A. Buffa, S. Corsaro, and A. Murli, Eds., Springer-Verlag (2003) 785-794. | Zbl
[12] , and, Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM J. Numer. Anal. 42 (2004) 434-459. | Zbl
[13] , and, Mixed discontinuous Galerkin approximation of the Maxwell operator: Non-stabilized formulation. J. Sci. Comput. 22 (2005) 325-356. | Zbl
[14] ,, and, Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math. 100 (2005) 485-518. | Zbl
[15] and, A posteriori error estimation for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374-2399. | Zbl
[16] and, Problèmes aux Limites Non-Homogènes et Applications. Dunod, Paris (1968). | Zbl
[17] , A finite element method for approximating the time-harmonic Maxwell equations. Numer. Math. 63 (1992) 243-261. | Zbl
[18] , Finite element methods for Maxwell's equations. Oxford University Press, New York (2003). | Zbl
[19] , A simple proof of convergence for an edge element discretization of Maxwell's equations, in Computational electromagnetics, C. Carstensen, S. Funken, W. Hackbusch, R. Hoppe and P. Monk, Eds., Springer-Verlag, Lect. Notes Comput. Sci. Engrg. 28 (2003) 127-141. | Zbl
[20] , A new family of mixed finite elements in . Numer. Math. 50 (1986) 57-81. | Zbl
[21] and, The -local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations. Math. Comput. 72 (2003) 1179-1214. | Zbl
[22] , and, Stabilized interior penalty methods for the time-harmonic Maxwell equations. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4675-4697. | Zbl
[23] , An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp. 28 (1974) 959-962. | Zbl
[24] and, -adaptive finite elements in electromagnetics. Comput. Methods Appl. Mech. Engrg. 169 (1999) 331-344. | Zbl
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