Skip to main content
SpringerLink
Log in

A stabilized finite element method on nonaffine grids for time-harmonic Maxwell’s equations

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

A stabilized mixed finite element method is proposed for solving the time-harmonic Maxwell’s equations, with the divergence constraint imposed by the multiplier in a weak sense. By a grad-div stabilization, for some lowest-order edge elements on nonaffine quadrilateral, hexahedral and prismatic grids, we prove a type of uniform convergence for the zero-frequency Maxwell’s equations, then prove the well-posedness and the convergence for the time-harmonic Maxwell’s equations. Numerical results confirm the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Nédéléc’s two papers are titled with ‘mixed finite elements’.

References

  1. Ambartsumyan, I., Khattatov, E., Lee, J.J., Yotov, I.: Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra. Math. Models Methods Appl. Sci. 29, 1037–1077 (2019)

    MathSciNet  Google Scholar 

  2. Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998)

    MathSciNet  Google Scholar 

  3. Arbogast, T., Correa, M.R.: Two families of H(div) mixed finite elements on quadrilaterals of minimal dimension. SIAM J. Numer. Anal. 54, 3332–3356 (2016)

    MathSciNet  Google Scholar 

  4. Arbogast, T., Tao, Z.: Construction of H(div)-conforming mixed finite elements on cuboidal hexahedra. Numer. Math. 142, 1–32 (2019)

    MathSciNet  Google Scholar 

  5. Arnold, D.N., Boffi, D., Falk, R.S.: Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42, 2429–2451 (2005)

    MathSciNet  Google Scholar 

  6. Bergot, M., Duruflé, M.: High-order optimal edge elements for pyramids, prisms and hexahedra. J. Comput. Phys. 232, 189–213 (2013)

    MathSciNet  Google Scholar 

  7. Bergot, M., Duruflé, M.: Approximation of H(div) with high-order optimal finite elements for pyramids, prisms and hexahedra. Comm. Comput. Phys. 14, 1372–1414 (2013)

    MathSciNet  Google Scholar 

  8. Bermúdez, A., Gamallo, P., Nogueiras, M.R., Rodríguez, R.: Approximation properties of lowest-order hexahedral Raviart-Thomas finite elements. C. R. Acad. Sci. Paris Ser. I 340, 687–692 (2005)

    MathSciNet  Google Scholar 

  9. Bernardi, C., Girault, V.: A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35, 1893–1916 (1998)

    MathSciNet  Google Scholar 

  10. Bonito, A., Guermond, J.-L., Luddens, F.: An interior penalty method with \(C^{0}\) finite elements for the approximation of the Maxwell equations in heterogeneous media: convergence analysis with minimal regularity, ESAIM: M2AN Math. Model. Numer. Anal. 50, 1457–1489 (2016)

    Google Scholar 

  11. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991)

    Google Scholar 

  12. Chen, W.B., Wang, Y.Q.: Minimal degree H(curl) and H(div) conforming finite elements on polytopal meshes. Math. Comp. 86, 2053–2087 (2017)

    MathSciNet  Google Scholar 

  13. Chen, W.B., Wang, Y.Q.: \(H^1\), H(curl) and H(div) conforming elements on polygon-based prisms and cones. Numer. Math. 145, 973–1004 (2020)

    MathSciNet  Google Scholar 

  14. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    Google Scholar 

  15. Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151, 221–276 (2000)

    MathSciNet  Google Scholar 

  16. Costabel, M., Dauge, M., Nicaise, S.: Singularities of Maxwell interface problems. ESAIM: M2AN Math. Model. Numer. Anal. 33, 627–649 (1999)

    MathSciNet  Google Scholar 

  17. de Frutos, J., Garcí-Archilla, B., John, V., Novo, J.: Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements. Adv. Comput. Math. 44, 195–225 (2018)

    MathSciNet  Google Scholar 

  18. Dillon, B.M., Liu, P.T.S., Webb, J.P.: Spurious modes in quadrilateral and triangular edge elements. COMPEL 13, 311–316 (1994)

    Google Scholar 

  19. Duan, H.Y., Lin, P., Saikrishnan, P., Tan, Roger C. E.: L2-projected least squares finite element methods for the Stokes equations. SIAM J. Numer. Anal. 44, 732–752 (2006)

    MathSciNet  Google Scholar 

  20. Duan, H.Y., Lin, P., Saikrishnan, P., Tan, Roger C. E.: A least squares finite element method for the magnetostatic problem in a multiply-connected Lipschitz domain. SIAM J. Numer. Anal. 45, 2537–2563 (2007)

    MathSciNet  Google Scholar 

  21. Duan, H.Y., Li, S., Tan, R.C.E., Zheng, W.Y.: A delta-regularization finite element method for a double curl problem with divergence-free constraint. SIAM J. Numer. Anal. 50, 3208–3230 (2012)

    MathSciNet  Google Scholar 

  22. Duan, H.Y., Hsieh, P.-W., Tan, Roger C.E., Yang, S.-Y.: Analysis of the small viscosity and large reaction coefficient in the computation of the generalized Stokes problem by a novel stabilized finite element method. Comput. Methods Appl. Mech. Eng. 271, 23–47 (2014)

    MathSciNet  Google Scholar 

  23. Duan, H.Y., Ma, J.H., Zou, J.: Mixed finite element method with Gauss’ law enforced for Maxwell eigenproblem. SIAM. J. Sci. Comput. 43, A3677–A3712 (2021)

    MathSciNet  Google Scholar 

  24. Dubach, E., Luce, R., Thomas, J.-M.: Pseudo-conforming polynomial Lagrange finite elements on quadrilaterals and hexahedra. Comm. Pure Appl. Anal. 8, 237–254 (2009)

    Google Scholar 

  25. Falk, R.S., Gatto, P., Monk, P.: Hexhedral H(div) and H(curl) finite elements. ESAIM: M2AN Math. Modell. Numer. Anal. 45, 115–143 (2011)

    Google Scholar 

  26. Fernandes, P., Gilardi, G.: Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Meth. Appl. Sci. 7, 957–991 (1997)

    MathSciNet  Google Scholar 

  27. Fernandes, P., Raffetto, M.: Counterexamples to currently accepted explanation for spurious modes and necessary and sufficient conditions to avoid them. IEEE Trans. Magnet. 38, 653–656 (2002)

    Google Scholar 

  28. Fiordilino, J.A., Layton, W., Rong, Y.: An efficient and modular grad-div stabilization. Comput. Methods Appl. Mech. Eng. 335, 327–346 (2018)

    MathSciNet  Google Scholar 

  29. Franca, L.P., Hughes, T.J.R.: Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Eng. 69, 89–129 (1988)

    MathSciNet  Google Scholar 

  30. Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin (1986)

    Google Scholar 

  31. Ingram, R., Wheeler, M., Yotov, I.: A multipoint flux mixed finite element method on hexahedra. SIAM J. Numer. Anal. 48, 1281–1312 (2010)

    MathSciNet  Google Scholar 

  32. Kikuchi, F.: Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism. Comput. Methods Appl. Mech. Eng. 64, 509–521 (1987)

    MathSciNet  Google Scholar 

  33. Kikuchi, F.: On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36, 479–490 (1989)

    MathSciNet  Google Scholar 

  34. Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003)

    Google Scholar 

  35. Mur, G., de Hoop, A.T.: A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media. IEEE Trans. Magnet. 21, 2188–2191 (1985)

    Google Scholar 

  36. Nédélec, J.-C.: Mixed finite elements in R3. Numer. Math. 35, 315–341 (1980)

    MathSciNet  Google Scholar 

  37. Nédélec, J.-C.: A new family of mixed finite elements in R3. Numer. Math. 50, 57–81 (1986)

    MathSciNet  Google Scholar 

  38. Nicaise, S.: Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal. 39, 784–816 (2002)

    MathSciNet  Google Scholar 

  39. Olshanskii, M.A., Reusken, A.: Grad-div stabilization for Stokes equations. Math. Comp. 73, 1699–1718 (2004)

    MathSciNet  Google Scholar 

  40. Olshanskii, M., Lube, G., Heister, T., Löwe, J.: Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 198, 3975–3988 (2009)

    MathSciNet  Google Scholar 

  41. Sboui, A., Jaffré, J., Roberts, J.: A composite mixed finite element for hexahedral grids. SIAM J. Sci. Comput. 31, 2623–2645 (2009)

    MathSciNet  Google Scholar 

  42. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54, 483–493 (1990)

    MathSciNet  Google Scholar 

  43. Wheeler, M., Xue, G.R., Yotov, I.: A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra. Numer. Math. 121, 165–204 (2012)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees’ valuable suggestions which have greatly helped us improve the presentation of this paper.

Funding

The authors are supported by National Natural Science Foundation of China (12261160361, 12371371, 11971366).

Author information

Authors and Affiliations

  1. School of Natural Sciences, Wuhan University of Technology, Wuhan, 430070, China

    Zhijie Du

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Huoyuan Duan

Authors
  1. Zhijie Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huoyuan Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Huoyuan Duan.

Ethics declarations

Competing interests

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Communicated by Gunilla Kreiss.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Du, Z., Duan, H. A stabilized finite element method on nonaffine grids for time-harmonic Maxwell’s equations. Bit Numer Math 63, 47 (2023). https://doi.org/10.1007/s10543-023-00988-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10543-023-00988-6

Keywords

Mathematics Subject Classification

Search

Navigation