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

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Large-Scale Scientific Computing (LSSC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7116))

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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.

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References

  1. Ciarlet, P.G.: Discrete maximum principle for finite-difference operators. Aequationes Math. 4, 338–352 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. AMS (1997)

    Google Scholar 

  4. Faragó, I.: Numerical Treatment of Linear Parabolic Problems. Dissertation for the degree MTA Doktora (2008)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  6. Ishihara, K.: Strong and weak discrete maximum principles for matrices associated with elliptic problems. Linear Algebra Appl. 88-89, 431–448 (1987)

    Article  MathSciNet  Google Scholar 

  7. Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, New York (2003)

    MATH  Google Scholar 

  8. Varga, R.S.: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs (1962)

    Google Scholar 

  9. Varga, R.: On discrete maximum principle. J. SIAM Numer. Anal. 3, 355–359 (1966)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Dep. of Appl. Anal. and Comp. Math., Eötvös Loránd University, Pázmány Péter sétány I/C, Budapest, H-1117, Hungary

    Miklós E. Mincsovics & Tamás L. Horváth

  2. Dep. of Math. and Comp. Sci., Széchenyi University, Egyetem tér 1, H-9026, Győr, Hungary

    Tamás L. Horváth

Authors
  1. Miklós E. Mincsovics
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  2. Tamás L. Horváth
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Editor information

Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev, Block 25A, 1113, Sofia, Bulgaria

    Ivan Lirkov  & Svetozar Margenov  & 

  2. Department of Informatics and Mathematical Modelling, Technical University of Denmark, Richard Petersens Plads, Building 321,, 2800, Kongens Lyngby, Denmark

    Jerzy Waśniewski

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Mincsovics, M.E., Horváth, T.L. (2012). On the Differences of the Discrete Weak and Strong Maximum Principles for Elliptic Operators. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2011. Lecture Notes in Computer Science, vol 7116. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29843-1_70

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  • DOI: https://doi.org/10.1007/978-3-642-29843-1_70

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-29842-4

  • Online ISBN: 978-3-642-29843-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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