Title: Double complexes and local cochain projections

Authors: Richard S. Falk and Ragnar Winther

Source: Numerical Methods Partial Differential Equations 31 (2015), 
        no. 2, 541-551

Abstract: The construction of projection operators which commute with the
exterior derivative and at the same time are bounded in the proper Sobolev
spaces, represents a key took in the recent stability analysis of finite
element exterior calculus.   The so-called bounded cochain projections have
been constructed by combining a smoothing operator and the unbounded canonical
projections defined by the degrees of freedom.  However, an undesired property
of these bounded projections is that, in contrast to the canonical
projections, they are nonlocal.  The purpose of this paper is to discuss a
recent alterntive construction of bounded cochain projections, which also
are local.  A key tool for the new construction is the structure of a double
complex, resembling the Cech-de Rham double complex of algebraic topology.

Keywords: cochain projections, finite element exterior calculus,
stability analysis

URL: http://www.math.rutgers.edu/~falk/papers/double-complexes-online.pdf