×

Innovative mimetic discretizations for electromagnetic problems. (English) Zbl 1191.78056

Summary: We introduce a discretization methodology for Maxwell equations based on Mimetic Finite Differences (MFD). Following the lines of the recent advances in MFD techniques [see F. Brezzi et al., Comput. Methods Appl. Mech. Eng. 196, No. 37–40, 3682–3692 (2007; Zbl 1173.76370)] and the references therein) and using some of the results of [F. Brezzi and A. Buffa, Scalar products of discrete differential forms, in preparation], we propose mimetic discretizations for several formulations of electromagnetic problems both at low and high frequency in the time-harmonic regime. The numerical analysis for some of the proposed discretizations has already been developed, whereas for others the convergence study is an object of ongoing research.

MSC:

78M20 Finite difference methods applied to problems in optics and electromagnetic theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
78A25 Electromagnetic theory (general)

Citations:

Zbl 1173.76370
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Teixeira, F. L.; Chew, W. C., Lattice electromagnetic theory from a topological viewpoint, J. Math. Phys., 40, 169-187 (1999) · Zbl 0968.78005
[2] Bossavit, A., Whitney forms: A class of finite elements for three dimensional computations in electromagnetism, IEE Proc., 135, 493-500 (1988)
[3] Mattiussi, C., An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology, J. Comput. Phys., 133, 289-309 (1997) · Zbl 0878.65091
[4] Boffi, D., Fortin operator and discrete compactness for edge elements, Numer. Math., 87, 229-246 (2000) · Zbl 0967.65106
[5] Monk, P., Finite element methods for Maxwell’s equations (2003), Oxford University Press: Oxford University Press New York · Zbl 1024.78009
[6] Boffi, D.; Fernandes, P.; Gastaldi, L.; Perugia, I., Computational models of electromagnetic resonators: Analysis of edge element approximation, SIAM J. Numer. Anal., 36, 1264-1290 (1999) · Zbl 1025.78014
[7] Brezzi, F.; Lipnikov, K.; Shashkov, M., Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal., 43, 1872-1896 (2005), (electronic) · Zbl 1108.65102
[8] Brezzi, F.; Buffa, A.; Lipnikov, K., Mimetic finite differences for elliptic problems, ESAIM: Math. Model. Numer. Anal., 43, 277-295 (2009) · Zbl 1177.65164
[9] Bochev, P. B.; Hyman, J. M., Principles of mimetic discretizations of differential operators, (Compatible Spatial Discretizations. Compatible Spatial Discretizations, IMA Vol. Math. Appl., vol. 142 (2006), Springer: Springer New York), 89-119 · Zbl 1110.65103
[10] Boffi, D., A note on the de Rham complex and a discrete compactness property, Appl. Math. Lett., 14, 33-38 (2001) · Zbl 0983.65125
[11] Demkowicz, L.; Buffa, A., \(H^1, H(curl)\) and \(H(div)\)-conforming projection-based interpolation in three dimensions. Quasi-optimal \(p\)-interpolation estimates, Comput. Methods Appl. Mech. Engrg., 194, 267-296 (2005) · Zbl 1143.78365
[13] Brezzi, F.; Lipnikov, K.; Simoncini, V., A family of mimetic finite difference methods on polygonal and polyhedral meshes, Math. Models Methods Appl. Sci., 15, 1533-1551 (2005) · Zbl 1083.65099
[14] Brezzi, F.; Lipnikov, K.; Shashkov, M.; Simoncini, V., A new discretization methodology for diffusion problems on generalized polyhedral meshes, Comput. Methods Appl. Mech. Engrg., 196, 3682-3692 (2007) · Zbl 1173.76370
[15] Amrouche, C.; Bernardi, C.; Dauge, M.; Girault, V., Vector potentials in three-dimensional non-smooth domains, Math. Meth. Appl. Sci., 21, 823-864 (1998) · Zbl 0914.35094
[16] Hyman, J. M.; Shashkov, M., Mimetic discretizations for Maxwells equations, J. Comput. Phys., 151, 881-909 (1999) · Zbl 0956.78015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.