Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Smoothed projections over weakly Lipschitz domains

Author: Martin W. Licht
Journal: Math. Comp. 88 (2019), 179-210
MSC (2010): Primary 65N30; Secondary 58A12
Published electronically: April 10, 2018
MathSciNet review: 3854055
Full-text PDF
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N30, 58A12

Retrieve articles in all journals with MSC (2010): 65N30, 58A12

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

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