Advertisement

Advances in Computational Mathematics

, Volume 45, Issue 2, pp 611–630 | Cite as

Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations

  • Chuanjun Chen
  • Kang Li
  • Yanping ChenEmail author
  • Yunqing Huang
Article
  • 104 Downloads

Abstract

In this paper, we present a second-order accurate Crank-Nicolson scheme for the two-grid finite element methods of the nonlinear Sobolev equations. This method involves solving a small nonlinear system on a coarse mesh with mesh size H and a linear system on a fine mesh with mesh size h, which can still maintain the asymptotically optimal accuracy compared with the standard finite element method. However, the two-grid scheme can reduce workload and save a lot of CPU time. The optimal error estimates in H1-norm show that the two-grid methods can achieve optimal convergence order when the mesh sizes satisfy h = O(H2). These estimates are shown to be uniform in time. Numerical results are provided to verify the theoretical estimates.

Keywords

Nonlinear Sobolev equations Two-grid finite element method Error estimates Crank-Nicolson scheme 

Mathematics Subject Classification (2010)

65N08 65N15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The authors thank the referees for valuable constructive comments and suggestions, which led to a significant improvement of this paper.

Funding information

The work is supported by the National Natural Science Foundation of China (Grant No. 11771375, 11671157, 11571297, 91430213), Shandong Province Natural Science Foundation(Grant No. ZR2018MAQ008), and China Postdoctoral Science Foundation funded project (Grant No. 2017M610501).

References

  1. 1.
    Axelsson, O., Layton, W.: A two-level discretization of nonlinear boundary value problems. SIAM J. Numer. Anal. 33, 2359–2374 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barenblatt, G.I., Zheltov, I.P., Kochina, I.N.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24, 1286–1303 (1960)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bi, C., Ginting, V.: Two-grid finite volume element method for linear and nonlinear elliptic problems. Numer. Math. 107, 177–198 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, C., Yang, M., Bi, C.: Two-grid methods for finite volume element approximations of nonlinear parabolic equations. J. Comput. Appl. Math. 228, 123–132 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, C., Liu, W.: Two-grid finite volume element methods for semilinear parabolic problems. Appl. Numer. Math. 60, 10–18 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, C., Liu, W.: A two-grid method for finite element solutions for nonlinear parabolic equations. Abstract and Applied Analysis 2012, 11 (2012). Article ID 391918MathSciNetzbMATHGoogle 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, P., Gurtin, M.E.: On a theory of heat conduction involving two temperatures. Zeitschrif für Angewandte Mathematik und Physik 19, 614–627 (1968)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, Y., Huang, Y.: A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations. Int. J. Numer. Methods Eng. 57, 193–209 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, Y., H. Liu, Liu, S.: Analysis of two-grid methods for reaction-diffusion equations by expanded mixed finite element methods. Int. J. Numer. Methods Eng. 69, 408–422 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, Y., Luan, P., Lu, Z.: Analysis of two-grid methods for nonlinear parabolic equations by expanded mixed finite element methods. Adv. Appl. Math. Mech. 1, 830–844 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, Y., Chen, L., Zhang, X.: Two-grid method for nonlinear parabolic equations by expanded mixed finite element methods. Numerical Methods for Partial Differential Equations 29, 1238–1256 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Davis, P.L.: A quasilinear parabolic and a related third order problem. J. Math. Anal. Appl. 40, 327–335 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dawson, C.N., Wheeler, M.F.: Two-grid methods for mixed finite element approximations of nonlinear parabolic equations. Contemp. Math. 180, 191–203 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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
  16. 16.
    Ewing, R.E: Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations. SIAM J. Numer. Anal. 15, 1125–1150 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gu, H.: Characteristic finite element methods for nonlinear Sobolev equations. Appl. Math. Comput. 102, 51–62 (1999)MathSciNetGoogle Scholar
  18. 18.
    Huang, Y., Chen, Y.: A multi-level iterative method for solving finite element equations of nonlinear singular two-point boundary value problems. Natural Science Journal of Xiantan University 16, 23–26 (1994)zbMATHGoogle Scholar
  19. 19.
    He, S., Li, H., Liu, Y.: Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations. Frontiers of Mathematics in China 8, 825–836 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lin, Y.: Galerkin methods for nonlinear Sobolev equations. Aequationes Math. 40, 54–66 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lin, Y., Zhang, T.: Finite element methods for nonlinear Sobolev equations with nonlinear boundary conditions. J. Math. Anal. Appl. 165, 180–191 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Liu, W.: A two-grid method for the semi-linear reaction-diffusion system of the solutes in the groundwater flow by finite volume element. Math. Comput. Simul. 142, 34–50 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ohm, M.R., Lee, H.Y.: L 2-error analysis of fully discrete discontinuous Galerkin approximations for nonlinear Sobolev equations. Bulletin of the Korean Mathematical Society 48, 897–915 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Shi, D., Tang, Q., Gong, W.: A low order characteristic-nonconforming finite element method for nonlinear Sobolev equation with convection-dominated term. Math. Comput. Simul. 114, 25–36 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Shi, D., Yan, F., Wang, J.: Unconditional superconvergence analysis of a new mixed finite element method for nonlinear Sobolev equation. Appl. Math. Comput. 274, 182–194 (2016)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Showalter, R.E: Existence and representation theorems for a semi-linear Sobolev equation in Banach space. SIAM J. Math. Anal. 3, 527–543 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sun, T., Yang, D.: A priori error estimates for interior penalty discontinuous Galerkin method applied to nonlinear Sobolev equations. Appl. Math. Comput. 200, 147–159 (2008)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Wu, L., Allen, M.B.: A two-grid method for mixed finite-element solution of reaction-diffusion equations. Numerical Methods for Partial Differential Equations 15, 317–332 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Xu, J.: A novel two-grid method for semi-linear equations. SIAM J. Sci. Comput. 15, 231–237 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Xu, J., Zhou, A: A two-grid discretization scheme for eigenvalue problems. Mathematics of Computation of the American Mathematical Society 70, 17–25 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Yan, J., Zhang, Q., Zhu, L., Zhang, Z: Two-grid methods for finite volume element approximations of nonlinear Sobolev equations. Numer. Funct. Anal. Optim. 37, 391–414 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhang, T., Lin, Y.P.: \(L^{\infty }\)-error bounds for some nonliner integro-differential equations by finite element approximations. Mathematica Numerica Sinica 13, 177–186 (1991)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Chuanjun Chen
    • 1
  • Kang Li
    • 1
    • 2
  • Yanping Chen
    • 3
    Email author
  • Yunqing Huang
    • 4
  1. 1.School of Mathematics and Information SciencesYantai UniversityYantaiChina
  2. 2.School of Data and Computer ScienceSun Yat-sen UniversityGuangzhouChina
  3. 3.School of Mathematical SciencesSouth China Normal UniversityGuangzhouChina
  4. 4.Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational ScienceXiangtan UniversityXiangtanChina

Personalised recommendations