Volume 55, Number 5, September-October 2021
|Page(s)||2075 - 2099|
|Published online||01 October 2021|
Local finite element approximation of Sobolev differential forms
Department of Mathematics, University of Hawaii at Manoa, Honolulu, USA
2 UCSD, Department of Mathematics, La Jolla 92093-0112, California, USA
3 ICERM, Brown University, Providence, 02903 Rhode Island, USA
* Corresponding author: email@example.com
Accepted: 8 July 2021
We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Clément interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.
Mathematics Subject Classification: 65N30
Key words: Broken Bramble-Hilbert lemma / finite element exterior calculus / Clément interpolant / Scott-Zhang interpolant
© The authors. Published by EDP Sciences, SMAI 2021
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