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Failure of the discrete maximum principle for an elliptic finite element problem

Domenii publicaţii > Matematica + Tipuri publicaţii > Articol în revistã ştiinţificã

Autori: Andrei Draganescu, Todd F. Dupont, and L. Ridgway Scott

Editorial: Mathematics of Computation, 74, p.1-23, 2005.


There has been a long-standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem. This is related to knowing whether the discrete Green’s function is positive for triangular meshes allowing sufficiently good approximation of $H^1$ functions. We study this question for the Poisson problem in two dimensions discretized via the Galerkin method with continuous piecewise linears. We give examples which show that in general the answer is negative, and furthermore we extend the number of cases where it is known to be positive. Our techniques utilize some new results about discrete Green’s functions that are of independent interest.

Cuvinte cheie: elemente finite, principiu discret de maxim // finite elements, Green's function, discrete maximum principle