Positivity preserving finite element approximation

We consider finite element operators defined on ``rough'' functions in a bounded polyhedron $\Omega$ in $\mathbb{R} ^N$. Insisting on preserving positivity in the approximations, we discover an intriguing and basic difference between approximating functions which vanish on the boundary of $\Omega$ and approximating general functions which do not. We give impossibility results for approximation of general functions to more than first order accuracy at extreme points of $\Omega$. We also give impossibility results about invariance of positive operators on finite element functions. This is in striking contrast to the well-studied case without positivity.