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Angle Conditions for Discrete Maximum Principles in Higher-Order FEM

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Numerical Mathematics and Advanced Applications 2009

Abstract

This contribution reviews the general theory of the discrete Green’s function and presents a numerical experiment indicating that the discrete maximum principle (DMP) fails to hold in the case of Poisson problem on any uniform triangulation of a triangular domain for orders of approximation three and higher. This extends the result [Computing 27, 145–154 (1981)] that the Laplace equation discretized by the higher-order FEM satisfies the DMP on a patch of triangular elements in exceptional cases only.

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References

  1. Brandts, J., Korotov, S., Křížek, M.: Dissection of the path-simplex in R n into n path-subsimplices. Linear Algebra Appl. 421, 382–393 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Christie, I., Hall, C.: The maximum principle for bilinear elements. Internat. J. Numer. Methods Engrg. 20, 549–553 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ciarlet, P.G.: Discrete variational Green’s function. I. Aequationes Math. 4, 74–82 (1970)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ciarlet, P.G., Varga, R.S.: Discrete variational Green’s function. II. One dimensional problem. Numer. Math. 16, 115–128 (1970)

    MATH  MathSciNet  Google Scholar 

  7. Drăgănescu, A., Dupont, T.F., Scott, L.R.: Failure of the discrete maximum principle for an elliptic finite element problem. Math. Comp. 74, 1–23 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Höhn, W., Mittelmann, H.-D.: Some remarks on the discrete maximum-principle for finite elements of higher order. Computing 27, 145–154 (1981)

    Article  MathSciNet  Google Scholar 

  9. Hopf, E.: Elementäre Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Sitzungsberichte Preussische Akademie der Wissenschaften, Berlin, 147–152 (1927)

    Google Scholar 

  10. Prestel, A., Delzell, C. N.: Positive polynomials: From Hilbert’s 17th problem to real algebra. Springer, Berlin (2001)

    MATH  Google Scholar 

  11. Šolín, P., Segeth, K., Doležel, I.: Higher-order finite element methods. Chapman & Hall/CRC, Boca Raton, FL (2004)

    MATH  Google Scholar 

  12. Vejchodský, T., Šolín, P.: Discrete maximum principle for higher-order finite elements in 1D. Math. Comp. 76, 1833–1846 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Vejchodský, T., Šolín, P.: Discrete maximum principle for a 1D problem with piecewise-constant coefficients solved by hp-FEM. J. Numer. Math. 15, 233–243 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Vejchodský, T., Šolín, P.: Discrete maximum principle for Poisson equation with mixed boundary conditions solved by hp-FEM. Adv. Appl. Math. Mech. 1, 201–214 (2009)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences, Žitná 25, CZ–11567, Prague 1, Czech Republic

    Tomáš Vejchodský

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  1. Tomáš Vejchodský
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Correspondence to Tomáš Vejchodský .

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Editors and Affiliations

  1. Dept. of Information Technology, Uppsala University, Uppsala, 751 05, Sweden

    Gunilla Kreiss

  2. Dept. of Information Technology, Uppsala University, Uppsala, 751 05, Sweden

    Per Lötstedt

  3. Dept. of Information Technology, Uppsala University, Uppsala, 751 05, Sweden

    Axel Målqvist

  4. Dept. of Information Technology, Uppsala University, Uppsala, 751 05, Sweden

    Maya Neytcheva

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Vejchodský, T. (2010). Angle Conditions for Discrete Maximum Principles in Higher-Order FEM. In: Kreiss, G., Lötstedt, P., Målqvist, A., Neytcheva, M. (eds) Numerical Mathematics and Advanced Applications 2009. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11795-4_97

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