Journal of Scientific Computing

, Volume 49, Issue 3, pp 311–331 | Cite as

Two-Grid Discontinuous Galerkin Method for Quasi-Linear Elliptic Problems

Article
First Online:
Received:
Revised:
Accepted:

Abstract

In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree r≥1 for a class of quasi-linear elliptic problems in Ω⊂ℝ2. We propose a two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasi-linear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasi-linear elliptic problem on a coarse space. Convergence estimates in a broken H 1-norm are derived to justify the efficiency of the proposed two-grid algorithm. Numerical experiments are provided to confirm our theoretical findings. As a byproduct of the technique used in the analysis, we derive the optimal pointwise error estimates of the SIPG method for the quasi-linear elliptic problems in ℝ d ,d=2,3 and use it to establish the convergence of the two-grid method for problems in Ω⊂ℝ3.

Keywords

Discontinuous Galerkin method SIPG Quasi-linear elliptic Two-grid algorithm Superconvergence 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Axelsson, O., Layton, W.: A two-level discretization of nonlinear boundary value problems. SIAM J. Numer. Anal. 33, 2359–2374 (1996) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Baker, G.A.: Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31, 45–59 (1977) MATHCrossRefGoogle Scholar
  5. 5.
    Bi, C., Ginting, V.: Two-grid finite volume element method for linear and nonlinear elliptic problems. Numer. Math. 108, 177–198 (2007) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Brenner, S.C.: Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Numer. Anal. 41, 306–324 (2003) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Brenner, S.C.: Discrete Sobolev and Poincaré inequalities for piecewise polynomial functions. Electron. Trans. Numer. Anal. 18, 42–48 (2004) MathSciNetMATHGoogle Scholar
  8. 8.
    Brenner, S.C., Scott, R.: The Mathematical Theory of Finite Element Methods. Springer, Berlin (1994) MATHGoogle Scholar
  9. 9.
    Brenner, S.C., Sung, L.-Y.: Multigrid algorithms for C 0 interior penalty methods. SIAM J. Numer. Anal. 44, 199–223 (2006) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Brenner, S.C., Zhao, J.: Convergence of multigrid algorithms for interior penalty methods. Appl. Numer. Anal. Comput. Math. 2, 3–18 (2004) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Brezzi, F., Manzini, G., Marini, D., Pietra, P., Russo, A.: Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equ. 16, 365–378 (2000) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Bustinza, R., Gatica, G.N., Cockburn, B.: An a posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems. J. Sci. Comput. 22–23, 147–185 (2005) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, H., Chen, Z.: Stability and convergence of mixed discontinuous finite element methods for second order differential problems. J. Numer. Math. 11, 253–287 (2003) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Chen, Z., Chen, H.: Pointwise error estimates of discontinuous Galerkin methods with penalty for second order elliptic problems. SIAM J. Numer. Anal. 42, 1146–1166 (2004) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978) MATHGoogle Scholar
  16. 16.
    Cockburn, B., Dong, B., Guzmán, J.: A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems. J. Sci. Comput. 40, 141–187 (2009) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Cockburn, B., Karniadakis, G., Shu, C.-W.: Discontinuous Galerkin Methods. Theory, Computation and Applications. Lect. Notes Comput. Sci. Eng., vol. 11. Springer, Berlin (2000) MATHCrossRefGoogle Scholar
  18. 18.
    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) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Douglas, J., Jr., Dupont, T.: A Galerkin method for a nonlinear Dirichlet problem. Math. Comput. 29, 689–696 (1975) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Douglas, J., Jr., Dupont, T., Serrin, J.: Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form. Arch. Ration. Mech. Anal. 42, 157–168 (1971) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Georgoulis, E.H., Houston, P.: Discontinuous Galerkin methods for the biharmonic problem. IMA J. Numer. Anal. 29, 573–594 (2009) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Gudi, T., Pani, A.K.: Discontinuous Galerkin methods for quasi-linear elliptic problems of nonmonotone type. SIAM J. Numer. Anal. 45, 163–192 (2007) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Gudi, T., Nataraj, N., Pani, A.K.: hp-Discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems. Numer. Math. 109, 233–268 (2008) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer Texts in Applied Mathematics, vol. 54. Springer, New York (2008) MATHCrossRefGoogle Scholar
  25. 25.
    Houston, P., Robson, J., Süli, E.: Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case. IMA J. Numer. Anal. 25, 726–749 (2005) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Houston, P., Schwab, C., Süli, E.: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39, 2133–2163 (2002) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Houston, P., Süli, E., Wihler, T.P.: A posteriori error analysis of hp-version discontinuous Galerkin finite element methods for second-order quasi-linear elliptic problems. IMA J. Numer. Anal. 28, 245–273 (2008) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Kanschat, G., Rannacher, R.: Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems. J. Numer. Math. 10, 249–274 (2002) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximations of second-order elliptic problems. SIAM J. Numer. Anal. 41, 2374–2399 (2003) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Karakashian, O.A., Pascal, F.: Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems. SIAM J. Numer. Anal. 45, 641–665 (2007) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Levy, D., Shu, C.-W., Yan, J.: Local discontinuous Galerkin methods for nonlinear dispersive equations. J. Comput. Phys. 196, 751–772 (2004) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Lovadina, C., Marini, L.D.: A-posteriori error estimates for discontinuous Galerkin approximations of second order elliptic problems. J. Sci. Comput. 40, 340–359 (2009) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Marion, M., Xu, J.: Error estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J. Numer. Anal. 32, 1170–1184 (1995) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Mozolevski, I., Süli, E., Bösing, P.R.: hp-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation. J. Sci. Comput. 30, 465–491 (2007) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Oden, J.T., Babuška, I., Baumann, C.E.: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146, 491–519 (1998) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Ortner, C., Süli, E.: Discontinuous Galerkin finite element approximation of nonlinear second-order elliptic and hyperbolic systems. SIAM J. Numer. Anal. 45, 1370–1397 (2007) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973 Google Scholar
  38. 38.
    Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, Philadelphia (2008) MATHCrossRefGoogle Scholar
  39. 39.
    Rivière, B., Wheeler, M.F., Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39, 902–931 (2001) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Utnes, T.: Two-grid finite element formulations of the incompressible Navier-Stokes equation. Commun. Numer. Methods Eng. 34, 675–684 (1997) MathSciNetCrossRefGoogle Scholar
  41. 41.
    Wu, L., Allen, M.B.: Two-grid method for mixed finite-element solution of coupled reaction-diffusion systems. Numer. Methods Partial Differ. Equ. 15, 589–604 (1999) MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Xu, J.: A new class of iterative methods for nonselfadjoint or indefinite elliptic problems. SIAM J. Numer. Anal. 29, 303–319 (1992) MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Xu, J.: A novel two-grid method for semi-linear equations. SIAM J. Sci. Comput. 15, 231–237 (1994) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70, 17–25 (1999) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsYantai UniversityShandongChina
  2. 2.Department of MathematicsUniversity of WyomingLaramieUSA

Personalised recommendations