Hermite interpolation of nonsmooth functions preserving boundary conditions

This article is devoted to the construction of a Hermite-type regularization operator transforming functions that are not necessarily into globally finite-element functions that are piecewise polynomials. This regularization operator is a projection, it preserves appropriate first and second order polynomial traces, and it has approximation properties of optimal order. As an illustration, it is used to discretize a nonhomogeneous Navier-Stokes problem, with tangential boundary condition.