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A new expanded mixed finite element method for Kirchhoff type parabolic equation

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Abstract

A new expanded mixed finite element method is constructed for solving the Kirchhoff type parabolic equation. Compared with the traditional mixed element methods, this method can deal well with the case that the diffusion coefficient is relatively small, and the coefficient matrix is symmetric positive definite. The regularity of the solution of the Kirchhoff type parabolic equation is considered, and the stability of the semidiscrete scheme is analyzed. Based on backward Euler difference procedure in time, a fully discrete algorithm is provided, and then the existence and uniqueness of the solution are proved by Brouwer’s fixed point theorem. Meanwhile, the optimal error estimates in L2-norm and H(div)-norm are derived. Finally, some numerical examples are presented to verify the effectiveness of the proposed algorithms with two-grid technique.

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References

  1. Chipot, M., Valente, V., Caffarelli, G.V.: Remarks on a nonlocal problem involving the Dirichlet energy. Rendiconti del Seminario matematico della Universita di Padova 110(4), 199–220 (2003)

    MathSciNet  MATH  Google Scholar 

  2. Zheng, S., Chipot, M.: Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms. Asymptot. Anal. 45, 301–312 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Peradze, J.: A numerical algorithm for the nonlinear Kirchhoff string eqaution. Numer. Math. 102, 311–342 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gudi, T.: Finite element method for a nonlocal problem of Kirchhoff type. SIAM J. Numer. Anal. 50, 657–668 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dond, A., Pani, A.K.: A priori and a posteriori estimates of conforming and mixed FEM for a Kirchhoff equation of elliptic type. CMAM 17, 217–236 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arbogast, T., Wheeler, M.F., Yotov, I.: Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34, 828–852 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, Z.: Expanded mixed finite element methods for linear second-order elliptic problems i. ESAIM Math. Model. Numer. Anal. 32(2), 479–499 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Z.: Expanded mixed finite element methods for quasilinear second order elliptic problems ii. ESAIM Math. Model. Numer. Anal. 32(4), 501–520 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Song, H., Jiang, L., Chen, G.: Convergence analysis of hybrid expanded mixed finite element method for elliptic equations. Comput. Math. Appl. 68, 1205–1219 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, Z.: Expanded mixed finite element methods for the 2nd order parabolic problems. J. Shandong Normal University 20(4), 1–5 (2005)

    Google Scholar 

  11. Liu, Y., Li, H., Wen, Z.: Expanded mixed finite element method for a kind of two-order linear parabolic differential equation. Numer. Math. J. Chinese Universities 30(3), 234–249 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Wang, K., Wang, Q.: Expanded mixed finite element method for second order hyperbolic equations. Comput. Math. Appl. 78, 2560–2574 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, N., Lin, P., Gao, F.: An expanded mixed finite element method for two-dimensional Sobolev equations. J. Comput. Appl. Math. 348, 342–355 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ibragimov, A., Kieu, T.T.: An expanded mixed finite element method for generalized Forchheimer flows in porous media. Comput. Math. Appl. 72, 1467–1483 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ji, B., Zhang, J., Yu, Y., Liu, J., Guo, H.: A new family of expanded mixed finite element methods for reaction-diffusion equations. J. Appl. Math. Comput. 68, 2857–2875 (2022)

  16. Rui, H., Kim, S., Kim, S.D.: A remark on least-squares mixed element methods for reaction diffusion problems. J. Comput. Appl. Math. 202(2), 230–236 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rui, H., Kim, S.D., Kim, S.: Split least-squares finite element methods for linear and nonlinear parabolic problems. J. Comput. Appl. Math. 223(2), 938–952 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sharma, N., Khebchareon, M., Pani, A.K.: A priori error estimates of expanded mixed FEM for Kirchhoff type parabolic equation. Numer. Algo. 83, 125–147 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yang, D.: A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media. Numer. Methods Partial Differ. Equ. 17(3), 229–249 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, J., Yang, D.: A splitting positive definite mixed element method for second order hyperbolic equations. Numer. Methods Partial Differ. Equ. 25(3), 622–636 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang, J., Yang, D., Shen, S., Zhu, J.: A new, MMOCAA-MFE Method for compressible miscible displacement in porous media. Appl. Numer. Math. 80, 65–80 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, J.: A new combined characteristic mixed finite element method for compressible miscible displacement problem. Numer. Algo. 81, 1157–1179 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, J., Han, H.: A new discontinuous Galerkin mixed finite element method for compressible miscible displacement problem. Comput. Math. Appl. 80, 1714–1725 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang, Y., Zhang, J.: The splitting mixed element method for parabolic equation and its application in chemotaxis model. Appl. Math. Comput. 313, 287–300 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Zhang, J., Zhang, Y., Guo, H., Fu, H.: A mass-conservative characteristic splitting mixed finite element method for convection-dominated Sobolev equation. Math. Comput. Simul. 160, 180–191 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, J., Shen, X., Guo, H., Fu, H.: Characteristic splitting mixed finite element analysis of compressible wormhole propagation. Appl. Numer. Math. 147, 66–87 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, J., Han, H., Yu, Y., Liu, J.: A new two-grid mixed finite element analysis of semi-linear reaction-diffusion equation. Comput. Math. Appl. 92, 172–179 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  28. Han, H., Zhang, J., Ji, B., Yu, Y., Yu, Y.: A new symmetric mixed element method for semi-linear parabolic problem based on two-grid discretization. Comput. Math. Appl. 108(15), 206–215 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  29. Thomee, V.: Galerkin finite element methods for parabolic problems. Springer (1997)

  30. Kesavan, S.: Topics in functional analysis and application. New age international (P) Ltd. Publishers (2008)

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Acknowledgements

The authors would like to express their sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper.

Funding

This work was supported by the Fundamental Research Funds for the Central Universities (22CX03020A) and the Major Scientific and Technological Projects of CNPC(D2019-184-001).

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Authors and Affiliations

  1. Department of Applied Mathematics, China University of Petroleum, Qingdao, 266580, China

    Bingjie Ji, Jiansong Zhang, Yue Yu & Yun Yu

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  1. Bingjie Ji
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  2. Jiansong Zhang
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  4. Yun Yu
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Correspondence to Jiansong Zhang.

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Ji, B., Zhang, J., Yu, Y. et al. A new expanded mixed finite element method for Kirchhoff type parabolic equation. Numer Algor 92, 2405–2432 (2023). https://doi.org/10.1007/s11075-022-01396-7

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