×

zbMATH — the first resource for mathematics

A hybridizable discontinuous Galerkin method for electromagnetics with a view on subsurface applications. (English) Zbl 1442.78001
Summary: Two Hybridizable Discontinuous Galerkin (HDG) schemes for the solution of Maxwell’s equations in the time domain are presented. The first method is based on an electromagnetic diffusion equation, while the second is based on Faraday’s and Maxwell-Ampère’s laws. Both formulations include the diffusive term depending on the conductivity of the medium. The three-dimensional formulation of the electromagnetic diffusion equation in the framework of HDG methods, the introduction of the conduction current term and the choice of the electric field as hybrid variable in a mixed formulation are the key points of the current study. Numerical results are provided for validation purposes and convergence studies of spatial and temporal discretizations are carried out. The test cases include both simulation in dielectric and conductive media.
MSC:
78A25 Electromagnetic theory, general
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Yee, K., Numerical solution of inital boundary value problems involving maxwell’s equations in isotropic media, IEEE Trans. Antennas and Propagation, 14, 302-307 (1966) · Zbl 1155.78304
[2] Barucq, H.; Delaurens, F.; Hanouzet, B., Method of absorbing boundary conditions: Phenomena of error stabilization, SIAM J. Numer. Anal., 35, 3, 1113-1129 (1998), URL http://www.jstor.org/stable/2587124 · Zbl 0914.35135
[3] Haber, E.; Ascher, U. M.; Oldenburg, D. W., Inversion of 3D electromagnetic data in frequency and time domain using an inexact all-at-once approach, Geophysics, 69, 5, 1216-1228 (2004), arXiv:https://doi.org/10.1190/1.1801938
[4] Um, E. S.; Harris, J. M.; Alumbaugh, D. L., 3D time-domain simulation of electromagnetic diffusion phenomena: A finite-element electric-field approach, Geophysics, 75, 4, F115-F126 (2010), arXiv:https://doi.org/10.1190/1.3473694
[5] Nguyen, N.; Peraire, J.; Cockburn, B., Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations, J. Comput. Phys., 230, 19, 7151-7175 (2011), URL http://www.sciencedirect.com/science/article/pii/S0021999111003226 · Zbl 1230.78031
[6] Christophe, A.; Descombes, S.; Lanteri, S., An implicit hybridized discontinuous Galerkin method for the 3D time-domain Maxwell equations, Appl. Math. Comput., 319, 395-408 (2018), recent Advances in Computing. URL http://www.sciencedirect.com/science/article/pii/S0096300317302758 · Zbl 1426.78031
[7] Nédélec, J. C., Mixed finite elements in R3, Numer. Math., 35, 3, 315-341 (1980)
[8] Nédélec, J. C., A new family of mixed finite elements in R3, Numer. Math., 50, 1, 57-81 (1986) · Zbl 0625.65107
[9] Sun, D.; Manges, J.; Yuan, X.; Cendes, Z., Spurious modes in finite-element methods, IEEE Antennas Propag. Mag., 37, 5, 12-24 (1995)
[10] Cockburn, B.; Gopalakrishnan, J.; Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47, 2, 1319-1365 (2009), arXiv:https://doi.org/10.1137/070706616 · Zbl 1205.65312
[11] Yakovlev, S.; Moxey, D.; Kirby, R. M.; Sherwin, S. J., To CG or to HDG: A comparative study in 3D, J. Sci. Comput., 67, 1, 192-220 (2016) · Zbl 1339.65225
[12] Kronbichler, M.; Wall, W., A performance comparison of continuous and discontinuous Galerkin methods with fast multigrid solvers, SIAM J. Sci. Comput., 40, 5, A3423-A3448 (2018), arXiv:https://doi.org/10.1137/16M110455X · Zbl 1402.65163
[13] Wang, T.; Hohmann, G. W., A finite-difference, time-domain solution for three-dimensional electromagnetic modeling, Geophysics, 58, 6, 797-809 (1993), arXiv:https://doi.org/10.1190/1.1443465
[14] Commer, M.; Newman, G., A parallel finite-difference approach for 3D transient electromagnetic modeling with galvanic sources, Geophysics, 69, 5, 1192-1202 (2004), arXiv:https://doi.org/10.1190/1.1801936
[15] Unsworth, M. J.; Travis, B. J.; Chave, A. D., Electromagnetic induction by a finite electric dipole source over a 2-D earth, Geophysics, 58, 2, 198-214 (1993), arXiv:https://doi.org/10.1190/1.1443406
[16] Um, E.; Harris, J.; L. Alumbaugh, D., An iterative finite element time-domain method for simulating three-dimensional electromagnetic diffusion in earth, Geophys. J. Int., 190, 871-886 (2012)
[17] Um, E.; Commer, M.; A. Newman, G., Efficient pre-conditioned iterative solution strategies for the electromagnetic diffusion in the Earth: finite-element frequency-domain approach, Geophys. J. Int., 193, 1460-1473 (2013)
[18] Um, E.; Commer, M.; A. Newman, G.; Hoversten, G., Finite element modelling of transient electromagnetic fields near steel-cased wells, Geophys. J. Int., 202, 901-913 (2015)
[19] Jackson, J. D., Classical Electrodynamics (1999), Wiley: Wiley New York, NY, URL https://www.wiley.com/en-us/Classical+Electrodynamics%2C+3rd+Edition-p-9780471309321 · Zbl 0920.00012
[20] Jin, J., The Finite Element Method in Electromagnetics (2002), Wiley · Zbl 1001.78001
[21] Monk, P., Finite Element Methods for Maxwell’s Equations (2003), Oxford University Press: Oxford University Press Oxford · Zbl 1024.78009
[22] Rosen, J., Redundancy and superfluity for electromagnetic fields and potentials, Amer. J. Phys., 48, 12, 1071-1073 (1980), arXiv:https://doi.org/10.1119/1.12289
[23] Haber, E., A mixed finite element method for the solution of the magnetostatic problem with highly discontinuous coefficients in 3D, Comput. Geosci., 4, 4, 323-336 (2000) · Zbl 1017.78007
[24] Constable, S., Ten years of marine CSEM for hydrocarbon exploration, Geophysics, 75, 5, 75A67-75A81 (2010), arXiv:https://doi.org/10.1190/1.3483451
[25] Cockburn, B.; Gopalakrishnan, J.; Sayas, F.-J., A projection-based error analysis of hdg methods, Math. Comp., 79, 1351-1367 (2010) · Zbl 1197.65173
[26] Cockburn, B.; Dong, B.; Guzmán, J., A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comp., 77, 264, 1887-1916 (2008), URL http://www.jstor.org/stable/40234595 · Zbl 1198.65193
[27] Stanglmeier, M.; Nguyen, N.; Peraire, J.; Cockburn, B., An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation, Comput. Methods Appl. Mech. Engrg., 300, 748-769 (2016), URL http://www.sciencedirect.com/science/article/pii/S0045782515003941 · Zbl 1423.76280
[28] Giacomini, M.; Karkoulias, A.; Sevilla, R.; Huerta, A., A superconvergent HDG method for Stokes flow with strongly enforced symmetry of the stress tensor, J. Sci. Comput., 77, 1679-1702 (2018) · Zbl 1404.76162
[29] La Spina, A.; Giacomini, M.; Huerta, A., Hybrid coupling of CG and HDG discretizations based on nitsche’s method, Comput. Mech., 311-330 (2019)
[30] Li, X.; Demmel, J.; Gilbert, J.; Grigori, i. L.; Shao, M.; Yamazaki, I., SuperLU Users’ GuideTech. Rep. LBNL-44289 (1999), Lawrence Berkeley National Laboratory
[31] Li, X. S.; Demmel, J. W., SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems, ACM Trans. Math. Softw., 29, 2, 110-140 (2003) · Zbl 1068.90591
[32] Heroux, M. A.; Bartlett, R. A.; Howle, V. E.; Hoekstra, R. J.; Hu, J. J.; Kolda, T. G.; Lehoucq, R. B.; Long, K. R.; Pawlowski, R. P.; Phipps, E. T.; Salinger, A. G.; Thornquist, H. K.; Tuminaro, R. S.; Willenbring, J. M.; Williams, A.; Stanley, K. S., An overview of the Trilinos project, ACM Trans. Math. Software, 31, 3, 397-423 (2005), URL http://doi.acm.org/10.1145/1089014.1089021 · Zbl 1136.65354
[33] Oliphant, T., NumPy: A Guide to NumPy (2006), Trelgol Publishing: Trelgol Publishing USA, URL http://www.numpy.org/
[34] Kronbichler, M.; Schoeder, S.; Müller, C.; Wall, W. A., Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation, Internat. J. Numer. Methods Engrg., 106, 9, 712-739 (2016), arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.5137, URL https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5137 · Zbl 1352.76058
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.