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Preprint 2-2004

Discrete Minimum and Maximum Principles for Finite Element Approximations of Non-Monotone Elliptic Equations

Source file is available as :   Postscript Document

Author(s) : Andreas Unterreiter , Ansgar Juengel

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 2-2004

MSC 2000

65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
65N12 Stability and convergence of numerical methods

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.

Keywords : Finite Elements, Semilinear Elliptic Equations, Variational Principle, Positivity-Preserving Approximation, Stampacchia Truncation Method

Notes :
Das Manuskript ist zur Veröffentlichung eingereicht.

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