Abstract
A non-stabilized mixed discontinuous Galerkin method for the discretization of the Maxwell operator on simplicial meshes is studied. In contrast to the stabilized scheme introduced in Houston, Perugia and Schötzau, Siam J. Numer. Anal., 42: 434–459, 2004, the proposed formulation contains no normal-jump stabilization; instead, it is based on discontinuous mixed-order \((P^{\ell})^{3} - P^{\ell + 1}\) elements for the approximation of the unknowns. A priori error bounds in the energy norm are derived that show convergence rates of the order \({\cal O}(h^{\ell})\) in the mesh size h. The error analysis relies on suitable decompositions of discontinuous spaces and on stability properties of the underlying conforming spaces. The formulation is tested on a set of numerical examples in two space dimensions.
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References
A. Alonso A. Valli (1999) ArticleTitleAn optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations Math. Compat. 68 607–631 Occurrence Handle10.1090/S0025-5718-99-01013-3
D. N. Arnold F. Brezzi B. Cockburn L. D. Marini (2001) ArticleTitleUnified analysis of discontinuous Galerkin methods for elliptic problems SIAM J. Numer. Anal. 39 1749–1779 Occurrence Handle10.1137/S0036142901384162
F Brezzi M Fortin (1991) Mixed and hybrid finite element methods. In Springer Series in Computational Mathematics Springer-Verlag New York
F. Brezzi M. Fortin (2001) ArticleTitleA minimal stabilisation procedure for mixed finite\break element methods Numer. Math. 89 457–491
L. Demkowicz L. Vardapetyan (1998) ArticleTitleModeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements Comput. Meth. Appl. Mech. Eng. 152 103–124 Occurrence Handle10.1016/S0045-7825(97)00184-9
R. Hiptmair (2002) ArticleTitleFinite elements in computational electromagnetism Acta Numer. 11 237–339 Occurrence Handle10.1017/S0962492902000041
P. Houston I. Perugia D. Schötzau (2004) ArticleTitleMixed discontinuous Galerkin approximation of the Maxwell operator SIAM J. Numer. Anal. 42 434–459 Occurrence Handle10.1137/S003614290241790X
P Houston I Perugia D Schötzau (2003) hp-DGFEM for Maxwell’s equations F Brezzi A Buffa S Corsaro A Murli (Eds) Numerical Mathematics and Advanced Applications ENUMATH 2001 Springer-Verlag New york 785–794
Houston, P., Perugia, I., and Schötzau, D. (2004). Nonconforming Mixed Finite Element Approximations to Time-Harmonic Eddy Current Problems. Technical Report 2003/15, University of Leicester, Department of Mathematics. IEEE Trans. Mag. 40, 1268–1273.
O. A. Karakashian F. Pascal (2003) ArticleTitleA posteriori error estimation for a discontinuous Galerkin approximation of second order elliptic problems SIAM J. Numer. Anal. 41 2374–2399 Occurrence Handle10.1137/S0036142902405217
P Monk (2003) Finite Element Methods for Maxwell’s Equations Oxford University Press Oxford
J. C. Nédélec (1980) ArticleTitleMixed finite elements in \({\mathbb R}\) 3 Numer. Math. 35 315–341 Occurrence Handle10.1007/BF01396415
J. C. Nédélec (1982) ArticleTitleÉléments finis mixtes incompressibles pour l’équation de Stokes dans \({\mathbb R}\) 3 Numer. Math. 39 97–112
J. C. Nédélec (1986) ArticleTitleA new family of mixed finite elements in \({\mathbb R}\) 3 Numer. Math. 50 57–81 Occurrence Handle10.1007/BF01389668
I. Perugia D. Schötzau (2003) ArticleTitleThe hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations Math. Comput. 72 1179–1214 Occurrence Handle10.1090/S0025-5718-02-01471-0
I. Perugia D. Schötzau P. Monk (2002) ArticleTitleStabilized interior penalty methods for the time-harmonic Maxwell equations Comput. Meth. Appl. Mech. Eng. 191 4675–4697 Occurrence Handle10.1016/S0045-7825(02)00399-7
L. Vardapetyan L. Demkowicz (1999) ArticleTitle hp-adaptive finite elements in electromagnetics Comput. Meth. Appl. Mech. Eng. 169 331–344 Occurrence Handle10.1016/S0045-7825(98)00161-3
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Houston, P., Perugia, I. & Schötzau, D. Mixed Discontinuous Galerkin Approximation of the Maxwell Operator: Non-Stabilized Formulation. J Sci Comput 22, 315–346 (2005). https://doi.org/10.1007/s10915-004-4142-8
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DOI: https://doi.org/10.1007/s10915-004-4142-8