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Positivity preserving finite element approximation. (English) Zbl 1001.41011

In this paper, finite elements on quasi-uniform partitions into simplices are studied. In two dimensions, the partitions are the usual triangulations, but the results in this paper work in higher dimensions, where general simplices are used, as well. The particular question addressed is whether there are higher order accurate approximations by finite elements on such partitions if positivity preservation is required. The approximation is by a Lagrange finite element reproducing a finite element operator which is positive and bounded. There is an impossibility result on the approximation of general functions to higher order accuracy at extreme points of a domain.

MSC:

41A25 Rate of convergence, degree of approximation
41A36 Approximation by positive operators
65D05 Numerical interpolation
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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