On the Regularity of Time-Harmonic Maxwell Equations with Impedance Boundary Conditions

Expand
  • 1 LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
    2 School of Mathematical Science, University of Chinese Academy of Sciences, Beijing 100049, China

Received date: 2023-09-28

  Revised date: 2024-02-09

  Accepted date: 2024-02-10

  Online published: 2025-04-21

Abstract

In this paper, we prove the H2 regularity of the solution to the time-harmonic Maxwell equations with impedance boundary conditions on domains with a C2 boundary under minimum regularity assumptions on the source and boundary functions.

Cite this article

Zhiming Chen . On the Regularity of Time-Harmonic Maxwell Equations with Impedance Boundary Conditions[J]. Communications on Applied Mathematics and Computation, 2025 , 7(2) : 759 -770 . DOI: 10.1007/s42967-024-00386-x

References

1. Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998)
2. Birman, Sh.M., Solomyak, M.Z.: L2-theory of the Maxwell operator in arbitrary domains. Russ. Math. Sur. 43, 75–96 (1987)
3. Bonito, A., Demlow, A., Nochetto, R.H.: Finite element methods for the Laplace-Beltrami operator. Handb. Numer. Anal. 21, 1–103 (2020)
4. Buffa, A., Costabel, M., Sheen, D.: On traces for H(curl;) in Lipschitz domains. J. Math. Anal. Appl. 276, 845–867 (2002)
5. Chen, Z., Li, K., Xiang, X.: A high order unfitted finite element method for time-harmonic Maxwell interface problems. arXiv:2301.08944v1 (2023)
6. Costabel, M., Dauge, M., Nicaise, S.: Corner singularities and analytic regularity for linear elliptic systems. Part I: smooth domains (2010). https://hal.science/hal-00453934v1
7. Dziuk, G., Elliott, C.E.: Finite element methods for surface PDEs. Acta Numer. 22, 289–396 (2013)
8. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)
9. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)
10. Hiptmair, R., Moiola, A., Perugia, I.: Stability results for the time-harmonic Maxwell equations with impedance boundary conditions. Math. Models Methods Appl. Sci. 21, 2263–2287 (2011)
11. Hofmann, S., Mitrea, M., Taylor, M.: Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains. J. Geom. Anal. 17, 593–647 (2007)
12. Lu, P., Wang, Y., Xu, X.: Regularity results for the time-harmonic Maxwell equations with impedance boundary condition. arXiv:1804.07856v1 (2018)
13. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
14. Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)
15. Nicaise, S., Tomezyk, J.: The time-harmonic Maxwell equations with impedance boundary conditions in convex polyhedral domains. In: Langer, U., Pauly, D., Repin, S. (eds.) Maxwell’s Equations: Analysis and Numerics, pp. 285–340. De Gruyter, Berlin (2019)
Options
Outlines

/