Skip to main content
Springer Nature Link
Log in

Hodge Decomposition Methods for a Quad-Curl Problem on Planar Domains

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We develop and analyze \(P_k\) Lagrange finite element methods for a quad-curl problem on planar domains that is based on the Hodge decomposition of divergence-free vector fields. Numerical results that illustrate the performance of the finite element methods are also presented.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces \((\)Second Edition\()\). Academic Press, Amsterdam (2003)

    Google Scholar 

  2. Alonso, A., Fernandes, P., Valli, A.: Weak and strong formulations for the time-harmonic eddy-current problem in general multi-connected domains. Eur. J. Appl. Math. 14, 387–406 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alonso-Rodríguez, A., Valli, A., Vázquez-Hernández, R.: A formulation of the eddy current problem in the presence of electric ports. Numer. Math. 113, 643–672 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Assous, F., Michaeli, M.: Hodge decomposition to solve singular static Maxwell’s equations in a non-convex polygon. Appl. Numer. Math. 60, 432–441 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Babuška, I., Suri, M.: The \(h\)-\(p\) version of the finite element method with quasiuniform meshes. M2AN Math. Model. Numer. Anal. 21, 199–238 (1987)

    Article  MATH  Google Scholar 

  7. Biskamp, D.: Magnetic Reconnection in Plasmas. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

  8. Bramble, J.H.: A proof of the inf-sup condition for the Stokes equations on Lipschitz domains. Math. Models Methods Appl. Sci. 13, 361–371 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brenner, S.C., Cui, J., Nan, Z., Sung, L.-Y.: Hodge decomposition for divergence-free vector fields and two-dimensional Maxwell’s equations. Math. Comput. 81, 643–659 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brenner, S.C., Gedicke, J., Sung, L.-Y.: An adaptive \(P_1\) finite element method for two-dimensional Maxwell’s equations. J. Sci. Comput. 55, 738–754 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Brenner, S.C., Gedicke, J., Sung, L.-Y.: Hodge decomposition for two-dimensional time harmonic Maxwell’s equations\(:\) impedance boundary condition. Math. Methods Appl. Sci. 40, 370–390 (2017). doi:10.1002/mma.3398

    Article  MATH  MathSciNet  Google Scholar 

  12. Brenner, S.C., Gedicke, J., Sung, L.-Y.: An adaptive \({P_1}\) finite element method for two-dimensional transverse magnetic time harmonic Maxwell’s equations with general material properties and general boundary conditions. J. Sci. Comput. 68, 848–863 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  13. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods \((\)Third Edition\()\). Springer, New York (2008)

    Book  MATH  Google Scholar 

  14. Cakoni, F., Colton, D., Monk, P., Sun, J.: The inverse electromagnetic scattering problem for anisotropic media. Inverse Probl. 26, 074004 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chacón, L., Simakov, A.N., Zocco, A.: Steady-state properties of driven magnetic reconnection in 2D electron magnetohydrodynamics. Phys. Rev. Lett. 99, 235001 (2007)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  17. Cui, J.: Multigrid methods for two-dimensional Maxwell’s equations on graded meshes. J. Comput. Appl. Math. 255, 231–247 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  18. Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)

    MATH  Google Scholar 

  19. Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34, 441–463 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  20. Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  22. Grisvard, P.: Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985)

    MATH  Google Scholar 

  23. Hong, Q., Hu, J., Shu, S., Xu, J.: A discontinuous Galerkin method for the fourth-order curl problem. J. Comput. Math. 30, 565–578 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lax, P.D.: Functional Analysis. Wiley-Interscience, New York (2002)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  26. Monk, P., Sun, J.: Finite element methods for Maxwell’s transmission eigenvalues. SIAM J. Sci. Comput. 34, B247–B264 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Nečas, J.: Direct methods in the theory of elliptic equations. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  28. Nečas, J.: Equations aux Dérivées Partielles. Presse de l’Université Montréal, Montreal (1965)

    Google Scholar 

  29. Sun, J.: A mixed FEM for the quad-curl eigenvalue problem. Numer. Math. 132, 185–200 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  30. Yosida, K.: Functional Analysis Classics in Mathematics. Springer, Berlin (1995). Reprint of the sixth (1980) edition

  31. Zheng, B., Hu, Q., Xu, J.: A nonconforming finite element method for fourth order curl equations in \(\mathbb{R}^{3}\). Math. Comput. 80, 1871–1886 (2011)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work of the first and third authors was supported in part by the National Science Foundation under Grant Nos. DMS-13-19172 and DMS-16-20273. The work of the second author was supported in part by the National Science Foundation under Grant No. DMS-15-21555.

Author information

Authors and Affiliations

  1. Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803, USA

    Susanne C. Brenner & Li-yeng Sung

  2. Department of Mathematics Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Jiguang Sun

Authors
  1. Susanne C. Brenner
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jiguang Sun
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Li-yeng Sung
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Susanne C. Brenner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brenner, S.C., Sun, J. & Sung, Ly. Hodge Decomposition Methods for a Quad-Curl Problem on Planar Domains. J Sci Comput 73, 495–513 (2017). https://doi.org/10.1007/s10915-017-0449-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0449-0

Keywords

Mathematics Subject Classification