Inhalt des Dokuments
Preprint 2-2004
Discrete Minimum and Maximum Principles for Finite Element Approximations of Non-Monotone Elliptic Equations
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.