Abstract
The aim of this article is to investigate the well-posedness, stability of solutions to the time-dependent Maxwell’s equations for electric field in conductive media in continuous and discrete settings, and study convergence analysis of the employed numerical scheme. The situation we consider would represent a physical problem where a subdomain is emerged in a homogeneous medium, characterized by constant dielectric permittivity and conductivity functions. It is well known that in these homogeneous regions the solution to the Maxwell’s equations also solves the wave equation, which makes computations very efficient. In this way our problem can be considered as a coupling problem, for which we derive stability and convergence analysis. A number of numerical examples validate theoretical convergence rates of the proposed stabilized explicit finite element scheme.
Article PDF
Avoid common mistakes on your manuscript.
References
D. N. Arnold, F. Brezzi, B. Cockburn, L. D. Marini: Unified analysis for discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2022), 1749–1779.
M. Asadzadeh: An Introduction to Finite Element Methods for Differential Equations. John Wiley & Sons, Hoboken, 2021.
M. Asadzadeh, L. Beilina: Stability and convergence analysis of a domain decomposition FE/FD method for Maxwell’s equations in the time domain. Algorithms 15 (2022), Article ID 337, 22 pages.
M. Asadzadeh, L. Beilina: A stabilized P1 domain decomposition finite element method for time harmonic Maxwell’s equations. Math. Comput. Simul. 204 (2023), 556–574.
F. Assous, P. Degond, E. Heinze, P. A. Raviart, J. Segre: On finite-element method for solving the three-dimensional Maxwell equations. J. Comput. Phys. 109 (1993), 222–237.
S. Balay, W. D. Gropp, L. C. McInnes, B. F. Smith: PETSc: The Portable, Extensible Toolkit for Scientific Computation. Available at http://www.mcs.anl.gov/petsc/.
L. Baudouin, M. de Buhan, S. Ervedoza, A. Osses: Carleman-based reconstruction algorithm for the waves. SIAM J. Numer. Anal. 59 (2021), 998–1039.
L. Beilina: Energy estimates and numerical verification of the stabilized domain decomposition finite element/finite difference approach for time-dependent Maxwell’s system. Cent. Eur. J. Math. 11 (2013), 702–733.
L. Beilina: WavES: Wave Equations Solutions. Available at http://www.waves24.com/ (2017).
L. Beilina, M. V. Klibanov: Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems. Springer, New York, 2012.
L. Beilina, E. Lindström: An adaptive finite element/finite difference domain decomposition method for applications in microwave imaging. Electronics 11 (2022), Article ID 1359, 33 pages.
L. Beilina, E. Lindström: A posteriori error estimates and adaptive error control for permittivity reconstruction in conductive media. Gas Dynamics with Applications in Industry and Life Sciences. Springer Proceedings in Mathematics & Statistics 429. Springer, Cham, 2023, pp. 117–141.
L. Beilina, V. Ruas: An explicit P1 finite element scheme for Maxwell’s equations with constant permittivity in a boundary neighborhood. Available at https://arxiv.org/abs/1808.10720 (2018), 38 pages.
A.-S. Bonnet-Ben Dhia, C. Hazard, S. Lohrengel: A singular field method for the solution of Maxwell’s equations in polyhedral domains. SIAM J. Appl. Math. 59 (1999), 2028–2044.
S. C. Brenner, L. R. Scott: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15. Springer, New York, 1994.
P. Ciarlet, Jr.: Augmented formulations for solving Maxwell equations. Comput. Methods Appl. Mech. Eng. 194 (2005), 559–586.
P. Ciarlet, Jr., E. Jamelot: Continuous Galerkin methods for solving the time-dependent Maxwell equations in 3D geometries. J. Comput. Phys. 226 (2007), 1122–1135.
G. C. Cohen: Higher-Order Numerical Methods for Transient Wave Equations. Scientific Computation. Springer, Berlin, 2002.
M. Costabel, M. Dauge: Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151 (2000), 221–276.
M. Costabel, M. Dauge: Weighted regularization of Maxwell equations in polyhedral domains: A rehabilitation of nodal finite elements. Numer. Math. 93 (2002), 239–277.
A. Elmkies, P. Joly: Edge finite elements and mass lumping for Maxwell’s equations: The 2D case. C. R. Acad. Sci., Paris, Sér. I 324 (1997), 1287–1293. (In French.)
K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential Equations. Cambridge University Press, Cambridge, 1996.
A. Ern, J.-L. Guermond: Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions. Comput. Math. Appl. 75 (2018), 918–932.
L. C. Evans: Partial Differential Equations. Graduate Studies in Mathematics 19. AMS, Providence, 1998.
Y. G. Gleichmann, M. J. Grote: Adaptive spectral inversion for inverse medium problems. Inverse Probl. 39 (2023), Article ID 125007, 27 pages.
E. Jamelot: Résolution des équations de Maxwell avec des éléments finis de Galerkin continus: PhD Thesis. L’Ecole Polytechnique, Paris, 2005. (In French.)
B.-n. Jiang: The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Scientific Computation. Springer, Berlin, 1998.
B.-n. Jiang, J. Wu, L. A. Povinelli: The origin of spurious solutions in computational electromagnetics. J. Comput. Phys. 125 (1996), 104–123.
J.-M. Jin: The Finite Element Method in Electromagnetics. John Wiley, New York, 1993.
C. Johnson: Numerical Solutions of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge, 1987.
P. Joly: Variational methods for time-dependent wave propagation problems. Topics in Computational Wave Propagation. Lecture Notes in Computational Science and Engineering 31. Springer, Berlin, 2003, pp. 201–264.
M. Křížek, P. Neittaanmäki: Finite Element Approximation of Variational Problems and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics 50. Longman, Harlow, 1990.
M. Křížek, P. Neittaanmäki: Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications. Kluwer Academic, Dordrecht, 1996.
M. Lazebnik, et al.: A large-scale study of the ultrawideband microwave dielectric properties of normal, benign and malignant breast tissues obtained from cancer surgeries. Phys. Med. Biol. 52 (2007), Article ID 6093, 20 pages.
J. B. Malmberg, L. Beilina: An adaptive finite element method in quantitative reconstruction of small inclusions from limited observations. Appl. Math. Inf. Sci. 12 (2018), 1–19.
P. Monk: Finite Element Methods for Maxwell’s Equations. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2003.
P. B. Monk, A. K. Parrott: A dispersion analysis of finite element methods for Maxwell’s equations. SIAM J. Sci. Comput. 15 (1994), 916–937.
C.-D. Munz, P. Omnes, R. Schneider, E. Sonnendrücker, U. Voss: Divergence correction techniques for Maxwell solvers based on a hyperbolic model. J. Comput. Phys. 161 (2000), 484–511.
J.-C. Nedelec: Mixed finite elements in ℝ3. Numer. Math. 35 (1980), 315–341.
K. D. Paulsen, D. R. Lynch: Elimination of vector parasites in finite element Maxwell solutions. IEEE Trans. Microw. Theory Tech. 39 (1991), 395–404.
N. T. Thành, L. Beilina, M. V. Klibanov, M. A. Fiddy: Reconstruction of the refractive index from experimental backscattering data using a globally convergent inverse method. SIAM J. Sci. Comput. 36 (2014), B273–B293.
N. T. Thành, L. Beilina, M. V. Klibanov, M. A. Fiddy: Imaging of buried objects from experimental backscattering time-dependent measurements using a globally convergent inverse algorithm. SIAM J. Imaging Sci. 8 (2015), 757–786.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of both authors is supported by the Swedish Research Council grant VR 2018-03661. Open access funding provided by University of Gothenburg.
Rights and permissions
Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Lindström, E., Beilina, L. Energy norm error estimates and convergence analysis for a stabilized Maxwell’s equations in conductive media. Appl Math (2024). https://doi.org/10.21136/AM.2024.0248-23
Received:
Published:
DOI: https://doi.org/10.21136/AM.2024.0248-23
Keywords
- Maxwell’s equation
- finite element method
- stability
- a priori error analysis
- energy error estimate
- convergence analysis