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Numerical Algorithms

, Volume 70, Issue 1, pp 93–111 | Cite as

A two-grid algorithm based on expanded mixed element discretizations for strongly nonlinear elliptic equations

  • Wei Liu
  • Zhe Yin
  • Jin Li
Original Paper

Abstract

An expanded mixed element method is presented to solve a strongly nonlinear elliptic problem. Existence and uniqueness of approximation solution are analyzed. Error estimates in L q and H s norms are also obtained in this paper. To solve the resulting nonlinear system of equations efficiently, we use a two-grid algorithm to decompose the nonlinear system into a small nonlinear system on a coarse grid with mesh size H and a linear system on a fine grid with mesh size h. It’s shown that the approximation still achieves asymptotically optimal as long as the mesh sizes satisfy H=O(h 1/2). Some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed numerical algorithm.

Keywords

Two-grid method Nonlinear elliptic equations Expanded mixed finite element Error estimates 

Mathematics Subject Classifications (2010)

65N12 65N15 65N30 

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References

  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bi, C., Ginting, V.: Two-grid finite volume element for linear and nonlinear elliiptic problems. Numer. Math. 108, 177–198 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brezzi, F., Douglas, J., Fortin, M., Marini, L.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47, 217–235 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, Z.: Expanded mixed element methods for linear second-order elliptic problems(I). RAIRO Model. Math. Anal. Numer. 32, 479–499 (1998)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen, Z.: Expanded mixed element methods for quasilinear second-order elliptic problems(II). RAIRO Model. Math. Anal. Numer. 32, 501–520 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, Z.: BDM mixed methods for a nonlinear elliptic problem. J. Comput. Appl. Math. 53, 207–223 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, L., Chen, Y.: Two-Grid Method for Nonlinear Reaction-Diffusion Equations by Mixed Finite Element Methods. J. Sci. Comput. 49, 383–401 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, Z., Douglas, J.: Prismatic mixed finite elements for second order elliptic problems. Calcolo 26, 135–148 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, C., Liu, W., Bi, C.: A two-grid characteristic finite volume element method for semilinear advection-dominated diffusion equations. Numer. Methods Partial Differ. Equ. 29, 1543–1562 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Crandall, M. G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27, 1–67 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dawson, C.N., Wheeler, M.F., Woodward, C.S.: A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. 35, 435–452 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gatica, G., Heuer, N.: An expanded mixed finite element approach via a dual-dual formulation and the minimum residual method. J. Comput. Appl. Math. 132, 371–385 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kim, D., Park, E.J.: A priori and a posteriori analysis of mixed finite element methods for nonlinear elliptic equations. SIAM J. Numer. Anal. 48, 1186–1207 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Milner, F. A., Park, E.J.: Mixed finite-element methods for Hamilton-Jacobi- Bellman-type equations. IMA J. Numer. Anal. 16, 401–414 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Park, E.J.: Mixed finite element methods for nonlinear second-order elliptic problems. SIAM J. Numer. Anal. 32, 865–885 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Milner, F.A.: Mixed finite element methods for quasilinear second order elliptic problems. Math. Comput. 44, 303–320 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Nedelec, J.C.: Mixed finite element in R 3. Numer. Math. 35, 315–341 (1980)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Proc. Conf. on Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., vol. 606, pp. 292–315. Springer, Berlin (1977)Google Scholar
  19. 19.
    Woodward, C.S., Dawson, C.N.: Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modelling flow into variably saturated porous media. SIAM J. Numer. Anal. 37, 701–724 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wu, L., Allen, M.B.: A two grid method for mixed element finite solution of reaction-diffusion equations. Numer. Methods Partial Differ. Equ. 15, 317–332 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Xu, J.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics and Statistics ScienceLudong UniversityYantaiPeople’s Republic of China
  2. 2.School of Mathematical SciencesShandong Normal UniversityJinanPeople’s Republic of China
  3. 3.School of ScienceShandong Jianzhu UniversityJinanPeople’s Republic of China

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