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

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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.

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Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Universität Mainz, Staudingerweg 9, 55099, Mainz, Germany

    Ansgar Jüngel

  2. Institut für Mathematik, MA 6-3, TU Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Andreas Unterreiter

Authors
  1. Ansgar Jüngel
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  2. Andreas Unterreiter
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Correspondence to Andreas Unterreiter.

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Mathematics Subject Classification (2000): 65N30, 65N12

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Jüngel, A., Unterreiter, A. Discrete minimum and maximum principles for finite element approximations of non-monotone elliptic equations. Numer. Math. 99, 485–508 (2005). https://doi.org/10.1007/s00211-004-0554-5

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  • DOI: https://doi.org/10.1007/s00211-004-0554-5

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