Local L2 Bounded projections in finite element exterior calculus

Finite element exterior calculus (FEEC) is an elegant framework that uses a complex of finite elements in any spatial dimension and can approximate solutions to the Hodge-Laplacian. The finite element complex on simplicial meshes can be thought of as generalizations of the Nédélec finite elements to higher dimensions and as higher order polynomial generalizations to the Whitney forms.  A chief tool in the analysis of FEEC for the Hodge-Laplacian are commuting bounded projections. There have been several such projections developed including the work of 1) Schoberl, 2) Christiansen and Winther, 3) Falk and Winther, 4)  Ern, Gudi, Smears,Vohralík to name a few. In this work we develop L2 bounded projections that are local.  This is joint work with Douglas Arnold.