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Smoothed projections over weakly Lipschitz domains


Author: Martin W. Licht
Journal: Math. Comp.
MSC (2010): Primary 65N30; Secondary 58A12
DOI: https://doi.org/10.1090/mcom/3329
Published electronically: April 10, 2018
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Abstract: We develop finite element exterior calculus over weakly Lipschitz domains. Specifically, we construct commuting projections from $ L^p$ de Rham complexes over weakly Lipschitz domains onto finite element de Rham complexes. The projections satisfy uniform bounds for finite element spaces with bounded polynomial degree over shape-regular families of triangulations. Thus we extend the theory of finite element differential forms to polyhedral domains that are weakly Lipschitz but not strongly Lipschitz. As new mathematical tools, we use the collar theorem in the Lipschitz category, and we show that the degrees of freedom in finite element exterior calculus are flat chains in the sense of geometric measure theory.


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Additional Information

Martin W. Licht
Affiliation: Department of Mathematics, University of California San Diego, 9500 Gilman Drive MC0112, La Jolla, California 92093-0112
Email: mlicht@ucsd.edu

DOI: https://doi.org/10.1090/mcom/3329
Keywords: Finite element exterior calculus, smoothed projection, weakly Lipschitz domain, Lipschitz collar, geometric measure theory
Received by editor(s): May 12, 2016
Received by editor(s) in revised form: April 21, 2017, and August 27, 2017
Published electronically: April 10, 2018
Additional Notes: This research was supported by the European Research Council through the FP7-IDEAS-ERC Starting Grant scheme, project 278011 STUCCOFIELDS
Article copyright: © Copyright 2018 American Mathematical Society

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