Energy norm error estimates and convergence analysis for a stabilized Maxwell's equations in conductive media

Eric Lindström; Larisa Beilina

Applications of Mathematics (2024)

  • Volume: 69, Issue: 4, page 415-436
  • ISSN: 0862-7940

Abstract

top
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.

How to cite

top

Lindström, Eric, and Beilina, Larisa. "Energy norm error estimates and convergence analysis for a stabilized Maxwell's equations in conductive media." Applications of Mathematics 69.4 (2024): 415-436. <http://eudml.org/doc/299599>.

@article{Lindström2024,
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.},
author = {Lindström, Eric, Beilina, Larisa},
journal = {Applications of Mathematics},
keywords = {Maxwell's equation; finite element method; stability; a priori error analysis; energy error estimate; convergence analysis},
language = {eng},
number = {4},
pages = {415-436},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Energy norm error estimates and convergence analysis for a stabilized Maxwell's equations in conductive media},
url = {http://eudml.org/doc/299599},
volume = {69},
year = {2024},
}

TY - JOUR
AU - Lindström, Eric
AU - Beilina, Larisa
TI - Energy norm error estimates and convergence analysis for a stabilized Maxwell's equations in conductive media
JO - Applications of Mathematics
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 4
SP - 415
EP - 436
AB - 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.
LA - eng
KW - Maxwell's equation; finite element method; stability; a priori error analysis; energy error estimate; convergence analysis
UR - http://eudml.org/doc/299599
ER -

References

top
  1. Arnold, D. N., Brezzi, F., Cockburn, B., Marini, L. D., 10.1137/S0036142901384162, SIAM J. Numer. Anal. 39 (2022), 1749-1779. (2022) Zbl1008.65080MR1885715DOI10.1137/S0036142901384162
  2. Asadzadeh, M., 10.1002/9781119671688, John Wiley & Sons, Hoboken (2021). (2021) Zbl1446.65001DOI10.1002/9781119671688
  3. Asadzadeh, M., Beilina, L., 10.3390/a15100337, Algorithms 15 (2022), Article ID 337, 22 pages. (2022) DOI10.3390/a15100337
  4. Asadzadeh, M., Beilina, L., 10.1016/j.matcom.2022.08.013, Math. Comput. Simul. 204 (2023), 556-574. (2023) Zbl07619073MR4484360DOI10.1016/j.matcom.2022.08.013
  5. Assous, F., Degond, P., Heinze, E., Raviart, P. A., Segre, J., 10.1006/jcph.1993.1214, J. Comput. Phys. 109 (1993), 222-237. (1993) Zbl0795.65087MR1253460DOI10.1006/jcph.1993.1214
  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., 10.1137/20M1315798, SIAM J. Numer. Anal. 59 (2021), 998-1039. (2021) Zbl1461.93213MR4244540DOI10.1137/20M1315798
  8. Beilina, L., 10.2478/s11533-013-0202-3, Cent. Eur. J. Math. 11 (2013), 702-733. (2013) Zbl1267.78044MR3015394DOI10.2478/s11533-013-0202-3
  9. Beilina, L., WavES: Wave Equations Solutions, Available at http://www.waves24.com/ (2017). (2017) 
  10. Beilina, L., Klibanov, M. V., 10.1007/978-1-4419-7805-9, Springer, New York (2012). (2012) Zbl1255.65168DOI10.1007/978-1-4419-7805-9
  11. Beilina, L., Lindström, E., 10.3390/electronics11091359, Electronics 11 (2022), Article ID 1359, 33 pages. (2022) DOI10.3390/electronics11091359
  12. Beilina, L., Lindström, E., 10.1007/978-3-031-35871-5_7, Gas Dynamics with Applications in Industry and Life Sciences Springer Proceedings in Mathematics & Statistics 429. Springer, Cham (2023), 117-141. (2023) MR4696623DOI10.1007/978-3-031-35871-5_7
  13. Beilina, L., Ruas, V., 10.48550/arXiv.1808.10720, Available at https://arxiv.org/abs/1808.10720 (2018), 38 pages. (2018) DOI10.48550/arXiv.1808.10720
  14. Dhia, A.-S. Bonnet-Ben, Hazard, C., Lohrengel, S., 10.1137/S00361399973233, SIAM J. Appl. Math. 59 (1999), 2028-2044. (1999) Zbl0933.78007MR1709795DOI10.1137/S00361399973233
  15. Brenner, S. C., Scott, L. R., 10.1007/978-1-4757-4338-8, Texts in Applied Mathematics 15. Springer, New York (1994). (1994) Zbl0804.65101MR1278258DOI10.1007/978-1-4757-4338-8
  16. P. Ciarlet, Jr., 10.1016/j.cma.2004.05.021, Comput. Methods Appl. Mech. Eng. 194 (2005), 559-586. (2005) Zbl1063.78018MR2105182DOI10.1016/j.cma.2004.05.021
  17. P. Ciarlet, Jr., E. Jamelot, 10.1016/j.jcp.2007.05.029, J. Comput. Phys. 226 (2007), 1122-1135. (2007) Zbl1128.78002MR2356870DOI10.1016/j.jcp.2007.05.029
  18. Cohen, G. C., 10.1007/978-3-662-04823-8, Scientific Computation. Springer, Berlin (2002). (2002) Zbl0985.65096MR1870851DOI10.1007/978-3-662-04823-8
  19. Costabel, M., Dauge, M., 10.1007/s002050050197, Arch. Ration. Mech. Anal. 151 (2000), 221-276. (2000) Zbl0968.35113MR1753704DOI10.1007/s002050050197
  20. Costabel, M., Dauge, M., 10.1007/s002110100388, Numer. Math. 93 (2002), 239-277. (2002) Zbl1019.78009MR1941397DOI10.1007/s002110100388
  21. Elmkies, A., Joly, P., 10.1016/S0764-4442(99)80415-7, C. R. Acad. Sci., Paris, Sér. I 324 (1997), 1287-1293 French. (1997) Zbl0877.65081MR1456303DOI10.1016/S0764-4442(99)80415-7
  22. Eriksson, K., Estep, D., Hansbo, P., Johnson, C., Computational Differential Equations, Cambridge University Press, Cambridge (1996). (1996) Zbl0946.65049MR1414897
  23. Ern, A., Guermond, J.-L., 10.1016/j.camwa.2017.10.017, Comput. Math. Appl. 75 (2018), 918-932. (2018) Zbl1409.65068MR3766493DOI10.1016/j.camwa.2017.10.017
  24. Evans, L. C., 10.1090/gsm/019, Graduate Studies in Mathematics 19. AMS, Providence (1998). (1998) Zbl0902.35002MR1625845DOI10.1090/gsm/019
  25. Gleichmann, Y. G., Grote, M. J., 10.1088/1361-6420/ad01d4, Inverse Probl. 39 (2023), Article ID 125007, 27 pages. (2023) Zbl07765707MR4664072DOI10.1088/1361-6420/ad01d4
  26. Jamelot, E., Résolution des équations de Maxwell avec des éléments finis de Galerkin continus: PhD Thesis, L'Ecole Polytechnique, Paris (2005), French. (2005) Zbl1185.65006
  27. Jiang, B.-n., 10.1007/978-3-662-03740-9, Scientific Computation. Springer, Berlin (1998). (1998) Zbl0904.76003MR1639101DOI10.1007/978-3-662-03740-9
  28. Jiang, B.-n., Wu, J., Povinelli, L. A., 10.1006/jcph.1996.0082, J. Comput. Phys. 125 (1996), 104-123. (1996) Zbl0848.65086MR1381806DOI10.1006/jcph.1996.0082
  29. Jin, J.-M., The Finite Element Method in Electromagnetics, John Wiley, New York (1993). (1993) Zbl0823.65124MR1903357
  30. Johnson, C., Numerical Solutions of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge (1987). (1987) Zbl0628.65098MR0925005
  31. Joly, P., 10.1007/978-3-642-55483-4_6, Topics in Computational Wave Propagation Lecture Notes in Computational Science and Engineering 31. Springer, Berlin (2003), 201-264. (2003) Zbl1049.78028MR2032871DOI10.1007/978-3-642-55483-4_6
  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). (1990) Zbl0708.65106MR1066462
  33. Křížek, M., Neittaanmäki, P., 10.1007/978-94-015-8672-6, Kluwer Academic, Dordrecht (1996). (1996) Zbl0859.65128MR1431889DOI10.1007/978-94-015-8672-6
  34. Lazebnik, M., al., et, 10.1088/0031-9155/52/20/002, Phys. Med. Biol. 52 (2007), Article ID 6093, 20 pages. (2007) DOI10.1088/0031-9155/52/20/002
  35. Malmberg, J. B., Beilina, L., 10.18576/amis/120101, Appl. Math. Inf. Sci. 12 (2018), 1-19. (2018) MR3747879DOI10.18576/amis/120101
  36. Monk, P., 10.1093/acprof:oso/9780198508885.001.0001, Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2003). (2003) Zbl1024.78009MR2059447DOI10.1093/acprof:oso/9780198508885.001.0001
  37. Monk, P. B., Parrott, A. K., 10.1137/0915055, SIAM J. Sci. Comput. 15 (1994), 916-937. (1994) Zbl0804.65122MR1278007DOI10.1137/0915055
  38. Munz, C.-D., Omnes, P., Schneider, R., Sonnendrücker, E., Voss, U., 10.1006/jcph.2000.6507, J. Comput. Phys. 161 (2000), 484-511. (2000) Zbl0970.78010MR1764247DOI10.1006/jcph.2000.6507
  39. Nedelec, J.-C., 10.1007/BF01396415, Numer. Math. 35 (1980), 315-341. (1980) Zbl0419.65069MR0592160DOI10.1007/BF01396415
  40. Paulsen, K. D., Lynch, D. R., 10.1109/22.75280, IEEE Trans. Microw. Theory Tech. 39 (1991), 395-404. (1991) DOI10.1109/22.75280
  41. Thành, N. T., Beilina, L., Klibanov, M. V., Fiddy, M. A., 10.1137/130924962, SIAM J. Sci. Comput. 36 (2014), B273--B293. (2014) Zbl1410.78018MR3199422DOI10.1137/130924962
  42. Thành, N. T., Beilina, L., Klibanov, M. V., Fiddy, M. A., 10.1137/140972469, SIAM J. Imaging Sci. 8 (2015), 757-786. (2015) Zbl1432.35259MR3327354DOI10.1137/140972469

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.