, 55:45 | Cite as

A unified framework for two-grid methods for a class of nonlinear problems

  • Dongho Kim
  • Eun-Jae ParkEmail author
  • Boyoon Seo


In this article, we present a unified error analysis of two-grid methods for a class of nonlinear problems. We first study the two-grid method of Xu by recasting the methodology in the abstract framework of Brezzi, Rappaz, and Raviart (BRR) for approximation of branches of nonsingular solutions and derive a priori error estimates. Our convergence results indicate that the correct scaling between fine and coarse meshes is given by \(h={{\mathcal {O}}}(H^2)\) for all the nonlinear problems which can be written in and applied to the BRR framework. Next, a correction step can be added to the two-grid algorithm, which allows the choice \(h={\mathcal O}(H^3)\). On the other hand, the particular BRR framework with duality pairing, if it is applied to a semilinear problem, allows a higher order relation \(h={{\mathcal {O}}}(H^4)\). Furthermore, even the choice \(h={{\mathcal {O}}}(H^5)\) is possible with the correction step either on fine mesh or coarse mesh. In addition, elliptic problems with gradient nonlinearities and the Naiver–Stokes equations are considered to illustrate our unified theory. Finally, numerical experiments are conducted to confirm our theoretical findings. Numerical results indicate that the correction step used as a simple postprocessing enhances the solution accuracy, particularly for problems with layers.


Two-grid finite element method Error estimates Nonlinear problems 

Mathematics Subject Classification

65N15 65N30 65N50 76D05 



The authors would like to express sincere thanks to the anonymous referee whose invaluable comments led to an improved version of the paper.


  1. 1.
    Bi, C., Ginting, V.: Two-grid finite volume element method for linear and nonlinear elliptic problems. Numer. Math. 107, 177–198 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bi, C., Ginting, V.: Two-grid discontinuous Galerkin method for quasi-linear elliptic problems. J. Sci. Comput. 49, 311–331 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)CrossRefGoogle Scholar
  4. 4.
    Brezzi, F., Rappaz, J., Raviart, P.A.: Finite dimensional approximation of nonlinear problems. Numer. Math. 36, 1–25 (1980)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Caloz, G., Rappaz, J.: Numerical analysis for nonlinear and bifurcation problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. V, pp. 487–637. North-Holland, Amsterdam (1997)Google Scholar
  6. 6.
    Chen, Y., Huang, Y., Yu, D.: A two-grid method for expanded mixed finite-element solution of semilinear reaction–diffusion equations. Intern. J. Numer. Methods Eng. 57, 193–209 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dawson, C., Wheeler, M.F.: Two-grid methods for mixed finite element approximations of nonlinear parabolic equations. In: Domain Decomposition Methods in Scientific and Engineering Computing (University Park, PA, 1993), 191.203, Contemp. Math., 180. American Mathematical Society, Providence, RI, (1994)Google Scholar
  8. 8.
    Dawson, C., Wheeler, M.F., Woodward, C.S.: A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. 35, 435–452 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Girault, V., Lions, J.-L.: Two-grid finite element schemes for the steady Navier–Stokes problem in polyhedra. Port. Math. 58, 25–57 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Girault, V., Lions, J.-L.: Two-grid finite element schemes for the transient Navier–Stokes. Math. Model. Numer. Anal. 35, 945–980 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations, pp. 27–28. Springer, New York (1986)CrossRefGoogle Scholar
  12. 12.
    Jin, J., Shu, S., Xu, J.: A two-grid discretization method for decoupling systems of partial differential equations. Math. Comput. 75(256), 1617–1626 (2006)MathSciNetCrossRefGoogle 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(3), 1186–1207 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kim, D., Park, E.-J., Seo, B.: Two-scale product approximation for semilinear parabolic problems in mixed methods. J. Korean Math. Soc. 51(2), 267–288 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Layton, W.: A two level discretization method for the Navier–Stokes equations. Comput. Math. Appl. 26, 33–38 (1993)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Layton, W., Lenferink, W.: Two-level Picard and modified Picard methods for Navier–Stokes equations. Appl. Math. Comput. 69, 263–274 (1995)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Layton, W., Tobiska, L.: A two-level method with backtracking for the Navier–Stokes equations. SIAM J. Numer. Anal. 35, 2035–2054 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Milner, F.A., Park, E.-J.: Mixed finite element methods for Hamilton–Jacobi–Bellman-type equations. IMA J. Numer. Anal. 16, 399–412 (1996)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shin, D.-W., Kang, Y., Park, E.-J.: \(C^0\)-discontinuous Galerkin methods for a wind-driven ocean circulation model: two-grid algorithm. Comput. Methods Appl. Mech. Eng. 328, 321–339 (2018)CrossRefGoogle Scholar
  20. 20.
    Wu, L., Allen, M., Park, E.-J.: Mixed finite-element solution of reaction–diffusion equations using a two-grid method. Comput. Methods Water Resour. XII, 217–224 (1998)Google Scholar
  21. 21.
    Wu, L., Allen, M.B.: A two-grid method for mixed finite-element solution of reaction–diffusion equations. Numer. Methods Part. Differ. Equ. 15(3), 317–332 (1999)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Xu, J.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33(5), 1759–1777 (1996)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhou, J., Hu, X., Zhong, L., Shu, S., Chen, L.: Two-grid methods for Maxwell eigenvalue problems. SIAM J. Numer. Anal. 52(4), 2027–2047 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche 2018

Authors and Affiliations

  1. 1.University CollegeYonsei UniversitySeoulKorea
  2. 2.Department of Computational Science and EngineeringYonsei UniversitySeoulKorea
  3. 3.Department of MathematicsYonsei UniversitySeoulKorea

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