Discrete minimum and maximum principles for finite element approximations of non-monotone elliptic equations

  • Authors:

  • Ansgar Jüngel;Andreas Unterreiter

  • Affiliations:

  • Universität Mainz, Fachbereich Mathematik und Informatik, Staudingerweg 9, 55099, Mainz, Germany;Institut für Mathematik, MA 6-3, TU Berlin, Fachbereich Mathematik und Informatik, Straße des 17. Juni 136, 10623, Berlin, Germany

  • Venue:
  • Numerische Mathematik
  • Year:
  • 2005

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Abstract

Uniform lower and upper bounds for positive finite-element approximations to semilinear elliptic equations in several space dimensions subject to mixed Dirichlet-Neumann boundary conditions are derived. The main feature is that the non-linearity may be non-monotone and unbounded. The discrete minimum principle provides a positivity-preserving approximation if the discretization parameter is small enough and if some structure conditions on the non-linearity and the triangulation are assumed. The discrete maximum principle also holds for degenerate diffusion coefficients. The proofs are based on Stampacchia’s truncation technique and on a variational formulation. Both methods are settled on careful estimates on the truncation operator.