Smoothed projections over weakly Lipschitz domains
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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.References
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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
- MR Author ID: 1225084
- Email: mlicht@ucsd.edu
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 179-210
- MSC (2010): Primary 65N30; Secondary 58A12
- DOI: https://doi.org/10.1090/mcom/3329
- MathSciNet review: 3854055