Positivity preserving finite element interpolation and applications to variational inequalities

Consider the classical problem of interpolating a rough (without point-values) function by continuous piecewise polynomials, but with the additional constraint of preserving positivity. We construct a positive interpolation operator into the space of piecewise linear finite elements with optimal approximation properties. We also give several intriguing impossibility results for such operators concerning boundary values at extreme points, invariance of finite element subspaces, and higher order accuracy. We finally apply positive interpolation to derive optimal a posteriori error estimators for elliptic variational inequalities both in the energy and maximum norms.

This work is joint with Z. Chen, K. Siebert, A. Vesser, and L. Wahlbin.