Journal of Applied Mathematics and Computing

, Volume 43, Issue 1–2, pp 249–269 | Cite as

A coupling method based on new MFE and FE for fourth-order parabolic equation

  • Yang LiuEmail author
  • Zhichao Fang
  • Hong Li
  • Siriguleng He
  • Wei Gao
Original Research


In this article, a coupling method of new mixed finite element (MFE) and finite element (FE) is proposed and analyzed for fourth-order parabolic partial differential equation. First, the fourth-order parabolic equation is split into the coupled system of second-order equations. Then, an equation is solved by finite element method, the other equation is approximated by the new mixed finite element method, whose flux belongs to the square integrable space replacing the classical H(div;Ω) space. The stability for fully discrete scheme is derived, and both semi-discrete and fully discrete error estimates are obtained. Moreover, the optimal a priori error estimates in L 2 and H 1-norm for both the scalar unknown u and the diffusion term γ and a priori error estimate in (L 2)2-norm for its flux σ are derived. Finally, some numerical results are provided to validate our theoretical analysis.


Fourth-order parabolic equation New coupling method New mixed scheme Finite element scheme Square integrable (L2(Ω))2 space Error estimates 

Mathematics Subject Classification

65M60 65N15 65N30 



The authors thank the anonymous referees and editors for their helpful comments and suggestions, which greatly improve the article. This work is supported by National Natural Science Fund (11061021), Natural Science Fund of Inner Mongolia Autonomous Region (2012MS0108, 2012MS0106, 2011BS0102), Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011, NJ10006, NJ10016), Program of Higher-level talents of Inner Mongolia University (125119, Z200901004, 30105-125132).


  1. 1.
    Li, J.C.: Full-order convergence of a mixed finite element method for fourth-order elliptic equations. J. Math. Anal. Appl. 230, 329–349 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Li, J.C.: Optimal error estimates of mixed finite element method for a fourth-order nonlinear elliptic problem. J. Math. Anal. Appl. 334, 183–195 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Li, J.C.: Optimal convergence analysis of mixed finite element methods for fourth-order elliptic and parabolic problems. Numer. Methods Partial Differ. Equ. 22, 884–896 (2006) CrossRefzbMATHGoogle Scholar
  4. 4.
    Li, J.C.: Mixed methods for fourth-order elliptic and parabolic problems using radial basis function. Adv. Comput. Math. 23, 21–30 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, Z.X.: Analysis of expanded mixed methods for fourth-order elliptic problems. Numer. Methods Partial Differ. Equ. 13(5), 483–503 (1997) CrossRefzbMATHGoogle Scholar
  6. 6.
    Christov, C.I., Pontes, J., Walgraef, D., Velarde, M.G.: Implicit time splitting for fourth-order parabolic equations. Comput. Methods Appl. Mech. Eng. 148, 209–224 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Barrett, J.W., Blowey, J.F., Garcke, H.: Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Numer. Math. 80, 525–556 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Zhang, T.: Finite element analysis for Cahn-Hilliard equation. Math. Numer. Sin. 28(3), 281–292 (2006) (in Chinese) MathSciNetGoogle Scholar
  9. 9.
    Danumjaya, P., Pani, A.K.: Mixed finite element methods for a fourth order reaction diffusion equation. Numer. Methods Partial Differ. Equ. 28(4), 1227–1251 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Liu, Y., Li, H., He, S., Gao, W., Fang, Z.C.: H 1-Galerkin mixed element method and numerical simulation for the fourth-order parabolic partial differential equations. Math. Numer. Sin. 34(2), 259–274 (2012) (in Chinese) MathSciNetzbMATHGoogle Scholar
  11. 11.
    King, B.B., Stein, O., Winkler, M.: A fourth-order parabolic equation modeling epitaxial thin film growth. J. Math. Anal. Appl. 286, 459–490 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    He, S., Li, H., Liu, Y.: Analysis of mixed finite element methods for fourth-order wave equations. Comput. Math. Appl. 65(1), 1–16 (2012) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, S.C., Chen, H.R.: New mixed element schemes for second order elliptic problem. Math. Numer. Sin. 32(2), 213–218 (2010) (in Chinese) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Shi, F., Yu, J.P., Li, K.T.: A new stabilized mixed finite-element method for Poisson equation based on two local Gauss integrations for linear element pair. Int. J. Comput. Math. 88(11), 2293–2305 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shi, D.Y., Zhang, Y.D.: High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equations. Appl. Math. Comput. 218(7), 3176–3186 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978) zbMATHGoogle Scholar
  17. 17.
    Luo, Z.D.: Mixed Finite Element Methods and Applications. China Sci. Press, Beijing (2006) (in Chinese) Google Scholar
  18. 18.
    Zwillinger, D.: Handbook of Differential Equations, 3rd edn. Academic Press, Boston (1997) Google Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2013

Authors and Affiliations

  • Yang Liu
    • 1
    Email author
  • Zhichao Fang
    • 1
  • Hong Li
    • 1
  • Siriguleng He
    • 1
  • Wei Gao
    • 1
  1. 1.School of Mathematical SciencesInner Mongolia UniversityHohhotChina

Personalised recommendations