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Keywords:
Maxwell's equation; finite element method; stability; a priori error analysis; energy error estimate; convergence analysis
Maxwell's equation; finite element method; stability; a priori error analysis; energy error estimate; convergence analysis
Summary:
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.
References:
[1] Arnold, D. N., Brezzi, F., Cockburn, B., Marini, L. D.: Unified analysis for discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2022), 1749-1779. DOI 10.1137/S0036142901384162 | MR 1885715 | Zbl 1008.65080
[2] Asadzadeh, M.: An Introduction to Finite Element Methods for Differential Equations. John Wiley & Sons, Hoboken (2021). DOI 10.1002/9781119671688 | Zbl 1446.65001
[3] Asadzadeh, M., Beilina, L.: 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. DOI 10.3390/a15100337
[4] Asadzadeh, M., Beilina, L.: A stabilized $P1$ domain decomposition finite element method for time harmonic Maxwell's equations. Math. Comput. Simul. 204 (2023), 556-574. DOI 10.1016/j.matcom.2022.08.013 | MR 4484360 | Zbl 07619073
[5] Assous, F., Degond, P., Heinze, E., Raviart, P. A., Segre, J.: On finite-element method for solving the three-dimensional Maxwell equations. J. Comput. Phys. 109 (1993), 222-237. DOI 10.1006/jcph.1993.1214 | MR 1253460 | Zbl 0795.65087
[6] Balay, S., Gropp, W. D., McInnes, L. C., Smith, B. F.: PETSc: The Portable, Extensible Toolkit for Scientific Computation. Available at http://www.mcs.anl.gov/petsc/
[7] Baudouin, L., Buhan, M. de, Ervedoza, S., Osses, A.: Carleman-based reconstruction algorithm for the waves. SIAM J. Numer. Anal. 59 (2021), 998-1039. DOI 10.1137/20M1315798 | MR 4244540 | Zbl 1461.93213
[8] Beilina, L.: 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. DOI 10.2478/s11533-013-0202-3 | MR 3015394 | Zbl 1267.78044
[9] Beilina, L.: WavES: Wave Equations Solutions. Available at http://www.waves24.com/ (2017).
[10] Beilina, L., Klibanov, M. V.: Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems. Springer, New York (2012). DOI 10.1007/978-1-4419-7805-9 | Zbl 1255.65168
[11] Beilina, L., Lindström, E.: An adaptive finite element/finite difference domain decomposition method for applications in microwave imaging. Electronics 11 (2022), Article ID 1359, 33 pages. DOI 10.3390/electronics11091359
[12] Beilina, L., Lindström, E.: 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), 117-141. DOI 10.1007/978-3-031-35871-5_7 | MR 4696623
[13] Beilina, L., Ruas, V.: 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. DOI 10.48550/arXiv.1808.10720
[14] Dhia, A.-S. Bonnet-Ben, Hazard, C., Lohrengel, S.: A singular field method for the solution of Maxwell's equations in polyhedral domains. SIAM J. Appl. Math. 59 (1999), 2028-2044. DOI 10.1137/S00361399973233 | MR 1709795 | Zbl 0933.78007
[15] Brenner, S. C., Scott, L. R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15. Springer, New York (1994). DOI 10.1007/978-1-4757-4338-8 | MR 1278258 | Zbl 0804.65101
[16] P. Ciarlet, Jr.: Augmented formulations for solving Maxwell equations. Comput. Methods Appl. Mech. Eng. 194 (2005), 559-586. DOI 10.1016/j.cma.2004.05.021 | MR 2105182 | Zbl 1063.78018
[17] 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. DOI 10.1016/j.jcp.2007.05.029 | MR 2356870 | Zbl 1128.78002
[18] Cohen, G. C.: Higher-Order Numerical Methods for Transient Wave Equations. Scientific Computation. Springer, Berlin (2002). DOI 10.1007/978-3-662-04823-8 | MR 1870851 | Zbl 0985.65096
[19] Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151 (2000), 221-276. DOI 10.1007/s002050050197 | MR 1753704 | Zbl 0968.35113
[20] Costabel, M., Dauge, M.: Weighted regularization of Maxwell equations in polyhedral domains: A rehabilitation of nodal finite elements. Numer. Math. 93 (2002), 239-277. DOI 10.1007/s002110100388 | MR 1941397 | Zbl 1019.78009
[21] Elmkies, A., Joly, P.: Edge finite elements and mass lumping for Maxwell's equations: The 2D case. C. R. Acad. Sci., Paris, Sér. I 324 (1997), 1287-1293 French. DOI 10.1016/S0764-4442(99)80415-7 | MR 1456303 | Zbl 0877.65081
[22] Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Computational Differential Equations. Cambridge University Press, Cambridge (1996). MR 1414897 | Zbl 0946.65049
[23] Ern, A., Guermond, J.-L.: Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions. Comput. Math. Appl. 75 (2018), 918-932. DOI 10.1016/j.camwa.2017.10.017 | MR 3766493 | Zbl 1409.65068
[24] Evans, L. C.: Partial Differential Equations. Graduate Studies in Mathematics 19. AMS, Providence (1998). DOI 10.1090/gsm/019 | MR 1625845 | Zbl 0902.35002
[25] Gleichmann, Y. G., Grote, M. J.: Adaptive spectral inversion for inverse medium problems. Inverse Probl. 39 (2023), Article ID 125007, 27 pages. DOI 10.1088/1361-6420/ad01d4 | MR 4664072 | Zbl 07765707
[26] Jamelot, E.: Résolution des équations de Maxwell avec des éléments finis de Galerkin continus: PhD Thesis. L'Ecole Polytechnique, Paris (2005), French. Zbl 1185.65006
[27] Jiang, B.-n.: The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Scientific Computation. Springer, Berlin (1998). DOI 10.1007/978-3-662-03740-9 | MR 1639101 | Zbl 0904.76003
[28] Jiang, B.-n., Wu, J., Povinelli, L. A.: The origin of spurious solutions in computational electromagnetics. J. Comput. Phys. 125 (1996), 104-123. DOI 10.1006/jcph.1996.0082 | MR 1381806 | Zbl 0848.65086
[29] Jin, J.-M.: The Finite Element Method in Electromagnetics. John Wiley, New York (1993). MR 1903357 | Zbl 0823.65124
[30] Johnson, C.: Numerical Solutions of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge (1987). MR 0925005 | Zbl 0628.65098
[31] Joly, P.: Variational methods for time-dependent wave propagation problems. Topics in Computational Wave Propagation Lecture Notes in Computational Science and Engineering 31. Springer, Berlin (2003), 201-264. DOI 10.1007/978-3-642-55483-4_6 | MR 2032871 | Zbl 1049.78028
[32] Křížek, M., Neittaanmäki, P.: Finite Element Approximation of Variational Problems and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics 50. Longman, Harlow (1990). MR 1066462 | Zbl 0708.65106
[33] Křížek, M., Neittaanmäki, P.: Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications. Kluwer Academic, Dordrecht (1996). DOI 10.1007/978-94-015-8672-6 | MR 1431889 | Zbl 0859.65128
[34] Lazebnik, M., al., et: 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. DOI 10.1088/0031-9155/52/20/002
[35] Malmberg, J. B., Beilina, L.: An adaptive finite element method in quantitative reconstruction of small inclusions from limited observations. Appl. Math. Inf. Sci. 12 (2018), 1-19. DOI 10.18576/amis/120101 | MR 3747879
[36] Monk, P.: Finite Element Methods for Maxwell's Equations. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2003). DOI 10.1093/acprof:oso/9780198508885.001.0001 | MR 2059447 | Zbl 1024.78009
[37] Monk, P. B., Parrott, A. K.: A dispersion analysis of finite element methods for Maxwell's equations. SIAM J. Sci. Comput. 15 (1994), 916-937. DOI 10.1137/0915055 | MR 1278007 | Zbl 0804.65122
[38] Munz, C.-D., Omnes, P., Schneider, R., Sonnendrücker, E., Voss, U.: Divergence correction techniques for Maxwell solvers based on a hyperbolic model. J. Comput. Phys. 161 (2000), 484-511. DOI 10.1006/jcph.2000.6507 | MR 1764247 | Zbl 0970.78010
[39] Nedelec, J.-C.: Mixed finite elements in $\Bbb R^3$. Numer. Math. 35 (1980), 315-341. DOI 10.1007/BF01396415 | MR 0592160 | Zbl 0419.65069
[40] Paulsen, K. D., Lynch, D. R.: Elimination of vector parasites in finite element Maxwell solutions. IEEE Trans. Microw. Theory Tech. 39 (1991), 395-404. DOI 10.1109/22.75280
[41] Thành, N. T., Beilina, L., Klibanov, M. V., Fiddy, M. A.: Reconstruction of the refractive index from experimental backscattering data using a globally convergent inverse method. SIAM J. Sci. Comput. 36 (2014), B273--B293. DOI 10.1137/130924962 | MR 3199422 | Zbl 1410.78018
[42] Thành, N. T., Beilina, L., Klibanov, M. V., Fiddy, M. A.: Imaging of buried objects from experimental backscattering time-dependent measurements using a globally convergent inverse algorithm. SIAM J. Imaging Sci. 8 (2015), 757-786. DOI 10.1137/140972469 | MR 3327354 | Zbl 1432.35259
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