Applications of Mathematics

, Volume 46, Issue 1, pp 13–28 | Cite as

About Delaunay Triangulations and Discrete Maximum Principles for the Linear Conforming FEM Applied to the Poisson Equation

  • Reiner Vanselow
Article

Abstract

The starting point of the analysis in this paper is the following situation: "In a bounded domain in ℝ2, let a finite set of points be given. A triangulation of that domain has to be found, whose vertices are the given points and which is `suitable' for the linear conforming Finite Element Method (FEM)." The result of this paper is that for the discrete Poisson equation and under some weak additional assumptions, only the use of Delaunay triangulations preserves the maximum principle.

linear conforming finite element method Delaunay triangulation discrete maximum principle 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    L. Angermann: An Introduction to Finite Volume Methods for Linear Elliptic Equations of Second Order. Preprint No. 164, Universität Erlangen-Nürnberg, Institut für Angewandte Mathematik I, 1995.Google Scholar
  2. [2]
    D. Braess: Finite Elemente. Springer-Verlag, Berlin, 1992.Google Scholar
  3. [3]
    F. Brezzi, M. Fortin: Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, 1991.Google Scholar
  4. [4]
    A. K. Cline, R. J. Renka: A constrained two-dimensional triangulation and the solution of closest node problems in the presence of barriers. SIAM J. Numer. Anal. 27 (1990), 1305–1321.Google Scholar
  5. [5]
    Ch. Großmann, H.-G. Roos: Numerik partieller Differentialgleichungen. Teubner, Stuttgart, 1992.Google Scholar
  6. [6]
    F. Kratsch, H.-G. Roos: Diskrete Maximumprinzipien und deren Anwendung. Preprint 07–02–87, TU Dresden, 1987.Google Scholar
  7. [7]
    F. P. Preparata, M. I. Shamos: Computational Geometry. An Introduction. Springer-Verlag, New York, 1985.Google Scholar
  8. [8]
    V. Ruas Santos: On the strong maximum principle for some piecewise linear finite element approximate problems of nonpositive type. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 473–491.Google Scholar
  9. [9]
    R. Vanselow: Relations between FEM and FVM applied to the Poisson equation. Computing 57 (1996), 93–104.Google Scholar
  10. [10]
    G. Windisch: M-Matrices in Numerical Analysis. Teubner, Leipzig, 1989.Google Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2001

Authors and Affiliations

  • Reiner Vanselow
    • 1
  1. 1.Institut fur Numerische MathematikTechnische Universitat Dresden, Mommsenstrasse 13DresdenGermany

Personalised recommendations