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

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Abstract

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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  4. Li, J.C.: Mixed methods for fourth-order elliptic and parabolic problems using radial basis function. Adv. Comput. Math. 23, 21–30 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Z.X.: Analysis of expanded mixed methods for fourth-order elliptic problems. Numer. Methods Partial Differ. Equ. 13(5), 483–503 (1997)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhang, T.: Finite element analysis for Cahn-Hilliard equation. Math. Numer. Sin. 28(3), 281–292 (2006) (in Chinese)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  17. Luo, Z.D.: Mixed Finite Element Methods and Applications. China Sci. Press, Beijing (2006) (in Chinese)

    Google Scholar 

  18. Zwillinger, D.: Handbook of Differential Equations, 3rd edn. Academic Press, Boston (1997)

    Google Scholar 

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Acknowledgements

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).

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Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Yang Liu, Zhichao Fang, Hong Li, Siriguleng He & Wei Gao

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  1. Yang Liu
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  2. Zhichao Fang
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  3. Hong Li
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  4. Siriguleng He
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  5. Wei Gao
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Correspondence to Yang Liu.

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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).

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Liu, Y., Fang, Z., Li, H. et al. A coupling method based on new MFE and FE for fourth-order parabolic equation. J. Appl. Math. Comput. 43, 249–269 (2013). https://doi.org/10.1007/s12190-013-0662-4

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  • DOI: https://doi.org/10.1007/s12190-013-0662-4

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