On the Differences of the Discrete Weak and Strong Maximum Principles for Elliptic Operators

  • Miklós E. Mincsovics
  • Tamás L. Horváth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7116)

Abstract

When choosing a numerical method to approximate the solution of a continuous mathematical problem, we need to consider which method results in an approximation that is not only close to the solution of the original problem, but possesses the important qualitative properties of the original problem, too. For linear elliptic problems the main qualitative properties are the various maximum principles. The preservation of the weak maximum principle was extensively investigated in the last decades, but not the strong maximum principle preservation. In this paper we focus on the latter property by giving its necessary and sufficient conditions, investigating the relation of the preservation of the strong and weak maximum principles and illustrating the differences between them with numerous examples.

Keywords

Maximum Principle Elliptic Operator Qualitative Property Simple Problem Strong Maximum Principle 
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References

  1. 1.
    Ciarlet, P.G.: Discrete maximum principle for finite-difference operators. Aequationes Math. 4, 338–352 (1970)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Draganescu, A., Dupont, T.F., Scott, L.R.: Failure of the discrete maximum principle for an elliptic finite element problem. Math. Comp. 74(249), 1–23 (2005)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. AMS (1997)Google Scholar
  4. 4.
    Faragó, I.: Numerical Treatment of Linear Parabolic Problems. Dissertation for the degree MTA Doktora (2008)Google Scholar
  5. 5.
    Hannukainen, A., Korotov, S., Vejchodský, T.: On Weakening Conditions for Discrete Maximum Principles for Linear Finite Element Schemes. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds.) NAA 2008. LNCS, vol. 5434, pp. 297–304. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Ishihara, K.: Strong and weak discrete maximum principles for matrices associated with elliptic problems. Linear Algebra Appl. 88-89, 431–448 (1987)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, New York (2003)MATHGoogle Scholar
  8. 8.
    Varga, R.S.: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs (1962)Google Scholar
  9. 9.
    Varga, R.: On discrete maximum principle. J. SIAM Numer. Anal. 3, 355–359 (1966)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Miklós E. Mincsovics
    • 1
  • Tamás L. Horváth
    • 1
    • 2
  1. 1.Dep. of Appl. Anal. and Comp. Math.Eötvös Loránd UniversityBudapestHungary
  2. 2.Dep. of Math. and Comp. Sci.Széchenyi UniversityGyőrHungary

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