BIT Numerical Mathematics

, Volume 59, Issue 4, pp 1093–1114 | Cite as

A curl-conforming weak Galerkin method for the quad-curl problem

  • Jiguang Sun
  • Qian ZhangEmail author
  • Zhimin Zhang


The quad-curl problem arises from the inverse electromagnetic scattering theory and magnetohydrodynamics. In this paper, a weak Galerkin method is proposed using the curl-conforming Nédélec elements. On one hand, the method avoids the construction of the curl–curl conforming elements and thus solves a smaller linear system. On the other hand, it is much simpler than the case of using the fully discontinuous elements. For polynomial spaces of order k, error estimates of \(O(h^{k-1})\) in the energy norm and of \(O(h^{k})\) in the \(H(\text {curl})\) norm are established, which are validated by the numerical examples.


Quad-curl problem WG methods Edge elements 

Mathematics Subject Classification

65N30 35Q60 65N15 35B45 



  1. 1.
    Brenner, S.C., Monk, P., Sun, J.: C\(^0\)IPG for the biharmonic eigenvalue problem. Spectral and high order methods for partial differential equations. Lect. Notes Comput. Sci. Eng. 106, 3–15 (2015)CrossRefGoogle Scholar
  2. 2.
    Brenner, S.C., Sun, J., Sung, L.: Hodge decomposition methods for a quad-curl problem on planar domains. J. Sci. Comput. 73(2–3), 495–513 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cakoni, F., Colton, D., Monk, P., Sun, J.: The inverse electromagnetic scattering problem for anisotropic media. Inverse Probl. 26(4), 074004 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cakoni, F., Haddar, H.: A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Probl. Imaging 1(3), 443–456 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, L., Huang, X.: Differential complexes, Helmholtz decompositions, and decoupling of mixed methods. (2016). arXiv preprint. arXiv:1611.03936
  6. 6.
    Du, Y., Zhang, Z.: A numerical analysis of the weak Galerkin method for the Helmholtz equation with high wave number. Commun. Comput. Phys. 22(1), 133–156 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hong, Q., Hu, J., Shu, S., Xu, J.: A discontinuous Galerkin method for the fourth-order curl problem. J. Comput. Math. 30(6), 565–578 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hong, Q., Wang, F., Wu, S., Xu, J.: A unified study of continuous and discontinuous Galerkin methods. Sci. China. Math. 62(1), 1–32 (2019)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kikuchi, F.: Mixed formulations for finite element analysis of magnetostatic and electrostatic problems. Japan J. Appl. Math. 6(2), 209–221 (1989)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)CrossRefGoogle Scholar
  12. 12.
    Monk, P., Sun, J.: Finite element methods for Maxwell’s transmission eigenvalues. SIAM J. Sci. Comput. 34, B247–B264 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mu, L., Wang, J., Ye, X., Zhang, S.: A \({C}^0\)-weak Galerkin finite element method for the biharmonic equation. J. Sci. Comput. 59(2), 473–495 (2012)CrossRefGoogle Scholar
  14. 14.
    Mu, L., Wang, J., Ye, X., Zhang, S.: A weak Galerkin finite element method for the Maxwell equations. J. Sci. Comput. 65, 363–386 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nédélec, J.C.: Mixed finite elements in \(\mathbb{R}^3\). Numer. Math. 35(1), 315–341 (1980)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nicaise, S.: Singularities of the quad-curl problem. J. Differ. Equations 264, 5025–5069 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sun, J.: A mixed FEM for the quad-curl eigenvalue problem. Numer. Math. 132(1), 185–200 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Sun, J., Xu, L.: Computation of Maxwell’s transmission eigenvalues and its applications in inverse medium problems. Inverse Probl. 29(10), 104013 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sun, J., Zhou, A.: Finite Element Methods for Eigenvalue Problems. Chapman and Hall/CRC, Boca Raton, FL (2016)CrossRefGoogle Scholar
  20. 20.
    Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241(15), 103–115 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second order elliptic problems. Math. Comp. 83(289), 2101–2126 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhai, Q., Xie, H., Zhang, R., Zhang, Z.: The weak Galerkin method for elliptic eigenvalue problems. Commun. Comput. Phys. 26(1), 160–191 (2019)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zhang, S.: Mixed schemes for quad-curl equations. ESAIM: M2AN 52(1), 147–161 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhang, S.: Regular decomposition and a framework of order reduced methods for fourth order problems. Numerische Mathematik 138, 241–271 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zheng, B., Xu, J.: A nonconforming finite element method for fourth order curl equations in \(\mathbb{R}^3\). Math. Comp. 80(276), 1871–1886 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesMichigan Technological UniversityHoughtonUSA
  2. 2.Beijing Computational Science Research CenterBeijingChina
  3. 3.Department of MathematicsWayne State UniversityDetroitUSA

Personalised recommendations