Double complexes and local cochain projections.
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 tool in the recent stability analysis of finite element exterior calculus. These 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 article is to discuss a recent alternative 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 Čech-de Rham double complex of algebraic topology.
Department of Mathematics, Rutgers University Piscataway, New Jersey, 08854.
Richard S
RS
Falk
Centre of Mathematics for Applications and Department of Mathematics, University of Oslo 0316, Oslo, Norway.
Ragnar
R
Winther
cochain projections; finite element exterior calculus; stability analysis