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Local bounded cochain projections

Authors: Richard S. Falk and Ragnar Winther
Journal: Math. Comp. 83 (2014), 2631-2656
MSC (2010): Primary 65N30
Published electronically: March 11, 2014
MathSciNet review: 3246803
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Abstract: We construct projections from $ H \Lambda ^k(\Omega )$, the space of differential $ k$ forms on $ \Omega $ which belong to $ L^2(\Omega )$ and whose exterior derivative also belongs to $ L^2(\Omega )$, to finite dimensional subspaces of $ H \Lambda ^k(\Omega )$ consisting of piecewise polynomial differential forms defined on a simplicial mesh of $ \Omega $. Thus, their definition requires less smoothness than assumed for the definition of the canonical interpolants based on the degrees of freedom. Moreover, these projections have the properties that they commute with the exterior derivative and are bounded in the $ H \Lambda ^k(\Omega )$ norm independent of the mesh size $ h$. Unlike some other recent work in this direction, the projections are also locally defined in the sense that they are defined by local operators on overlapping macroelements, in the spirit of the Clément interpolant. A double complex structure is introduced as a key tool to carry out the construction.

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Additional Information

Richard S. Falk
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

Ragnar Winther
Affiliation: Centre of Mathematics for Applications and Department of Mathematics, University of Oslo, 0316 Oslo, Norway

Keywords: Cochain projections, finite element exterior calculus
Received by editor(s): November 22, 2012
Received by editor(s) in revised form: April 15, 2013
Published electronically: March 11, 2014
Additional Notes: The work of the first author was supported in part by NSF grant DMS-0910540.
The work of the second author was supported by the Norwegian Research Council.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.