On Superlinear PCG Methods for FDM Discretizations of Convection-Diffusion Equations

  • János Karátson
  • Tamás Kurics
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5434)


The numerical solution of linear convection-diffusion equations is considered. Finite difference discretization leads to an algebraic system solved by a suitable preconditioned CG method, where the preconditioning approach is based on equivalent operators. Our goal is to study the superlinear convergence of the preconditioned CG iteration and to find mesh independent behaviour on a model problem. This is an analogue of previous results where FEM discretization was used.


Conjugate Gradient Method Superlinear Convergence Equivalent Operator Linear Convergence Coercivity Condition 
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.
    Ashby, S.F., Manteuffel, T.A., Saylor, P.E.: A taxonomy for conjugate gradient methods. SIAM J. Numer. Anal. 27(6), 1542–1568 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Axelsson, O.: A generalized conjugate gradient least square method. Numer. Math. 51, 209–227 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    Axelsson, O., Karátson, J.: Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators. Numer. Math. 99(2), 197–223 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Axelsson, O., Karátson, J.: Mesh independent superlinear PCG rates via compact-equivalent operators. SIAM J. Numer. Anal. 45(4), 1495–1516 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    D’yakonov, E.G.: On an iterative method for the solution of finite difference equations (in Russian). Dokl. Akad. Nauk SSSR 138, 522–525 (1961)MathSciNetGoogle Scholar
  7. 7.
    Faber, V., Manteuffel, T., Parter, S.V.: On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations. Adv. Appl. Math. 11, 109–163 (1990)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gunn, J.E.: The numerical solution of \(\nabla\cdot a \nabla u=f\) by a semi-explicit alternating direction iterative method. Numer. Math. 6, 181–184 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Karátson, J., Kurics, T.: Superlinearly convergent PCG algorithms for some nonsymmetric elliptic systems. J. Comp. Appl. Math. 212(2), 214–230 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Manteuffel, T., Otto, J.: Optimal equivalent preconditioners. SIAM J. Numer. Anal. 30, 790–812 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Manteuffel, T., Parter, S.V.: Preconditioning and boundary conditions. SIAM J. Numer. Anal. 27(3), 656–694 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • János Karátson
    • 1
  • Tamás Kurics
    • 1
  1. 1.Department of Applied Analysis and Computational MathematicsELTE UniversityBudapestHungary

Personalised recommendations