On Weakening Conditions for Discrete Maximum Principles for Linear Finite Element Schemes

  • Antti Hannukainen
  • Sergey Korotov
  • Tomáš Vejchodský
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5434)


In this work we discuss weakening requirements on the set of sufficient conditions due to Ph. Ciarlet [4,5] for matrices associated to linear finite element schemes, which is commonly used for proving validity of discrete maximum principles (DMPs) for the second order elliptic problems.


Dirichlet Boundary Condition Elliptic Problem Poisson Problem Basic Mesh Diagonal Dominance 
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.
    Bouchon, F.: Monotonicity of some perturbations of irreducibly diagonally dominant M-matrices. Numer. Math. 105, 591–601 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brandts, J., Korotov, S., Křížek, M.: The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem. Linear Algebra Appl. 429, 2344–2357 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bramble, J.H., Hubbard, B.E.: On a finite difference analogue of an elliptic boundary problem which is neither diagonally dominant nor of non-negative type. J. Math. and Phys. 43, 117–132 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ciarlet, P.G.: Discrete maximum principle for finite-difference operators. Aequationes Math. 4, 338–352 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ciarlet, P.G., Raviart, P.-A.: Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Engrg. 2, 17–31 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hannukainen, A., Korotov, S., Vejchodský, T.: Discrete maximum principle for FE-solutions of the diffusion-reaction problem on prismatic meshes. J. Comput. Appl. Math. (to appear)Google Scholar
  7. 7.
    Karátson, J., Korotov, S.: Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions. Numer. Math. 99, 669–698 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Karátson, J., Korotov, S., Křížek, M.: On discrete maximum principles for nonlinear elliptic problems. Math. Comput. Simulation 76, 99–108 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Korotov, S., Křížek, M., Neittaanmäki, P.: Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle. Math. Comp. 70, 107–119 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Křížek, M., Qun, L.: On diagonal dominance of stiffness matrices in 3D. East-West J. Numer. Math. 3, 59–69 (1995)zbMATHGoogle Scholar
  11. 11.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and quasilinear elliptic equations. Leon Ehrenpreis Academic Press, New York (1968)zbMATHGoogle Scholar
  12. 12.
    Ruas Santos, V.: On the strong maximum principle for some piecewise linear finite element approximate problems of non-positive type. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29, 473–491 (1982)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Varga, R.: Matrix Iterative Analysis. Prentice Hall, New Jersey (1962)Google Scholar
  14. 14.
    Varga, R.: On discrete maximum principle. J. SIAM Numer. Anal. 3, 355–359 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Vejchodský, T., Šolín, P.: Discrete maximum principle for higher-order finite elements in 1D. Math. Comp. 76, 1833–1846 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Antti Hannukainen
    • 1
  • Sergey Korotov
    • 1
  • Tomáš Vejchodský
    • 2
  1. 1.Institute of MathematicsHelsinki University of TechnologyEspooFinland
  2. 2.Institute of MathematicsCzech Academy of SciencesPrague 1Czech Republic

Personalised recommendations