A Two-Grid Method on Layer-Adapted Meshes for a Semilinear 2D Reaction-Diffusion Problem

  • Ivanka T. Angelova
  • Lubin G. Vulkov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5910)


A singularly perturbed semilinear reaction-diffusion equation, posed in the unit square, is discetizied by a two-grid scheme on layer-adapted meshes. In the first step, the nonlinear problem is discretized on coarse grid. In the second step, the problem is discretized on a fine grid and linearized around the interpolation of the computed solution on the first step. We show theoretically and numerically that the global error on Shishkin or Bakhvalov mesh is the same as would have been obtained if the nonlinear problem had been solved directly on the fine grid.


Coarse Grid Singularly Perturb Shishkin Mesh Piecewise Linear Basis Function Singularly Perturb Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Angelova, I.T., Vulkov, L.G.: Comparison of the Two-Grid Method on Different Meshes for a Singularly Perturbed Semilinear Problem. Amer. Inst. of Phys. 1067, 305–312 (2008)Google Scholar
  2. 2.
    Axelsson, O.: On Mesh Independence and Newton Methods. Applications of Math. 4-5, 249–265 (1993)MathSciNetGoogle Scholar
  3. 3.
    Bakhvalov, N.S.: On the Optimization Methods for Solcing Boundary Value Problems with Bounsary Layers. Zh. Vychisl. Math. Fiz. 24, 841–859 (1969) (in Russian)Google Scholar
  4. 4.
    Clavero, C., Gracia, J., O’Riordan, E.: A Parameter Robust Numerical Method for a Two Dimensionsl Reaction-Diffusion Problem. Math. Comp. 74, 1743–1758 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Duran, A., Lombardi, A.: Finite element approximation of convection-diffusion problems using graded meshes. Appl. Numer. Math. 56, 1314–1325 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Farrell, P.A., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: On the Non-Existence of ε-Uniform Finite Difference Method on Uniform Meshes for Semilinear Two-Point Boundary Value Problems. Math. Comp. 67(222), 603–617 (1999)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Gilbarg, D., Trudinger, N.: Ellipric Partial Differential Equations of Second Order. Springer, Heidelberg (1998)Google Scholar
  8. 8.
    Han, H., Kellog, R.B.: Differentiability Properties of Solutions of the Equation \(-\varepsilon ^{2} \triangle u +ru = f(x,y)\) in a Square. SIAM J. Math. Anal. 21, 394–408 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Herceg, D., Surla, K., Radeka, I., Malicic, I.: Numerical Experiments with Different Schemes for Singularly Perturbed Problem. Novi Sad J. Math. 31, 93–101 (2001)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Linss, T.: Layer-Adapted Meshex for Convection-Diffusion Problems. Habilitation, TU Dresden (2007)Google Scholar
  11. 11.
    Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Sinhular Perturbed Problems. World Scientific, Singapore (1996)Google Scholar
  12. 12.
    Roos, H., Stynes, M., Tobiska, L.: Numerical Methods for Singular Perturbed Differential Equations. In: Convection-Diffusion and Flow Problems. Springer, Berlin (2008)Google Scholar
  13. 13.
    Shishkin, G.I., Shishkina, L.P.: A High-Order Richardson Method for a Quadilinear Singularly Perturbed Elliptic Reaction-Diffusion Equation. Diff. Eqns. 41(7), 1030–1039 (2005) (in Russian)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Stynes, M., O’Riordan, E.: A Uniformly Concergent Galerkin Method on a Shishkin Mesh for a Convection-Diffjsion Problem. J. Math. Anal. Applic. 214, 36–54 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Vulanovic, R.: Finite-Difference Methods for a Class fo Strongly Nonlinear Singular Perturbation Problems. Num. Math. Theor. Appl. 1(2), 235–244 (2008)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Vulkov, L., Zadorin, A.: Two-Grid Interpolation Algorithms for Difference Schemes of Exponential Type for Semilinear Diffusion Convection-Dominated Equations. Amer. Inst. of Phys. 1067, 284–292 (2008)Google Scholar
  17. 17.
    Vulkov, L., Zadorin, A.: A Two-Grid Algorithm for Solution of the Difference Equations of a System, of Singularly Perturbed Semilinear Equations. LNCS, vol. 5434, pp. 582–589. Springer, Heidelberg (2009)Google Scholar
  18. 18.
    Xu, J.: A Novel Two-Grid Method for Semilinear Ellipptic Equations. SIAM, J. Sci. Comput. 15(1), 231–237 (1994)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Ivanka T. Angelova
    • 1
  • Lubin G. Vulkov
    • 1
  1. 1.Faculty of Natural Science and EducationUniversity of RousseRousseBulgaria

Personalised recommendations