Positivity preserving finite element approximation

We consider finite element operators defined on “rough” functions in a bounded polyhedron Ω in RN . Insisting on preserving positivity in the approximations, we discover an intriguing and basic difference between approximating functions which vanish on the boundary of Ω 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 Ω. 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.