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Multigrid Methods for a Mixed Finite Element Method of the Darcy–Forchheimer Model

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Abstract

An efficient nonlinear multigrid method for a mixed finite element method of the Darcy–Forchheimer model is constructed in this paper. A Peaceman–Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The nonlinear equation can be solved element-wise with a closed formulae. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number which controls the strength of the nonlinearity. By comparing the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computational cost.

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References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Arnold, D.N., Falk, R.S., Winther, R.: Preconditioning in \(H(\text{ div })\) and applications. Math. Comp. 66(219), 957–984 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, D.N., Falk, R.S., Winther, R.: Multigrid in \(H(\text{ div })\) and \(H(\text{ curl })\). Numer. Math. 85(2), 197–217 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aristotelous, A., Karakashian, O., Wise, S.: A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn–Hilliard equation and an efficient nonlinear multigrid solver. Discrete Contin. Dyn. Syst. Ser. B 18(9), 2211–2238 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science Publishers LTD, London (1979)

    Google Scholar 

  6. Bank, R.E., Douglas, C.C.: Sharp estimates for multigrid rates of convergence with general smoothing and acceleration. SIAM J. Numer. Anal. 22, 617–633 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. Braess, D., Sarazin, R.: An efficient smoother for the Stokes equation. Appl. Numer. Math. 23(1), 3–20 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comp. 31(138), 333–390 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  9. Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial, 2nd edn. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  10. Chen, L.: \(i\)FEM: an integrated finite element methods package in MATLAB. Technical report, University of California at Irvine (2009)

  11. Chen, L.: Multigrid methods for saddle point systems using constrained smoothers. Comput. Math. Appl. 70(12), 2854–2866 (2015)

    Article  MathSciNet  Google Scholar 

  12. Chen, L.: Multigrid methods for constrained minimization problems and application to saddle point problems. arXiv:1601.04091, pp. 1–27 (2016)

  13. Ciarlet, P.G.: The finite element method for elliptic problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  14. Forchheimer, P.: Wasserbewegung durch Boden. Z. Ver. Deutsch. Ing. 45, 1782–1788 (1901)

    Google Scholar 

  15. Girault, V., Wheeler, M.F.: Numerical discretization of a Darcy–Forchheimer model. Numer. Math. 110, 161–198 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hackbusch, W.: Multigrid Methods and Applications. Springer, Heidelberg (1985)

    Book  MATH  Google Scholar 

  17. López, H., Molina, B., Salas, J.J.: Comparison between different numerical discretizations for a Darcy–Forchheimer model. ETNA 34, 187–203 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Mardal, K., Winther, R.: Uniform preconditioners for the time dependent Stokes problem. Numer. Math. 98(2), 305–327 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Olshanskii, M., Peters, J., Reusken, A.: Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations. Numer. Math. 105(1), 159–191 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pan, H., Rui, H.: Mixed element method for two-dimensional Darcy–Forchheimer model. J. Sci. Comp. 52, 563–587 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Park, E.J.: Mixed finite element method for generalized Forchheimer flow in porous media. Numer. Methods Partial Differ. Equ. 21, 213–228 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Reusken, A.: Convergence of the multigrid full approximation scheme for a class of elliptic mildly nonlinear boundary value problems. Numer. Math. 52, 251–277 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rui, H., Liu, W.: A two-grid block-centered finite difference method for Darcy–Forchheimer flow in porous media. SIAM J. Numer. Anal. 53(4), 1941–1962 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rui, H., Pan, H.: A block-centered finite difference method for the Darcy–Forchheimer model. SIAM J. Numer. Anal. 50(5), 2612–2631 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rui, H., Zhao, D., Pan, H.: A block-centered finite difference method for Darcy–Forchheimer model with variable Forchheimer number. Numer. Methods Partial Differ. Equ. 31, 1603–1622 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ruth, D., Ma, H.: On the derivation of the Forchheimer equation by means of the averaging theorem. Transp Porous Media 7, 255–264 (1992)

    Article  Google Scholar 

  27. Salas, J.J., López, H., Molina, B.: An analysis of a mixed finite element method for a Darcy–Forchheimer model. Math. Comput. Model. 57, 2325–2338 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Tai, X.: Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities. Numer. Math. 93(4), 755–786 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tai, X., Xu, J.: Global and uniform convergence of subspace correction methods for some convex optimization problems. Math. Comput. 71(237), 105–125 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, Y., Rui, H.: Stabilized Crouzeix–Raviart element for Darcy–Forchheimer model. Numer. Methods Partial Differ. Equ. 31, 1568–1588 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wise, S.: Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn–Hilliard–Hele–Shaw system of equations. J. Sci. Comp. 44(1), 38–68 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yavneh, I., Dardyk, G.: A multilevel nonlinear method. SIAM J. Sci. Comp. 28(1), 24–46 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank the anonymous referee for the valuable suggestions and careful reading, which have helped us to improve the presentation.

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, Shandong, China

    Jian Huang & Hongxing Rui

  2. Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing, 100124, China

    Long Chen

  3. Department of Mathematics, University of California at Irvine, Irvine, CA, 92697, USA

    Long Chen

Authors
  1. Jian Huang
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  2. Long Chen
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  3. Hongxing Rui
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Corresponding author

Correspondence to Long Chen.

Additional information

The work of Jian Huang and Hongxing Rui was supported by the National Natural Science Foundation of China Grant No. 11671233, and in part by the Science Challenge Project No. JCKY2016212A502. Long Chen was supported by NSF Grant DMS-1418934, in part by NIH Grant P50GM76516, and in part by the Sea Poly Project of Beijing Overseas Talents. The work of Jian Huang was supported by 2014 China Scholarship Council (CSC).

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Huang, J., Chen, L. & Rui, H. Multigrid Methods for a Mixed Finite Element Method of the Darcy–Forchheimer Model. J Sci Comput 74, 396–411 (2018). https://doi.org/10.1007/s10915-017-0466-z

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  • DOI: https://doi.org/10.1007/s10915-017-0466-z

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