Hybridization and postprocessing in finite element exterior calculus
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- by Gerard Awanou, Maurice Fabien, Johnny Guzmán and Ari Stern;
- Math. Comp. 92 (2023), 79-115
- DOI: https://doi.org/10.1090/mcom/3743
- Published electronically: September 2, 2022
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Abstract:
We hybridize the methods of finite element exterior calculus for the Hodge–Laplace problem on differential $k$-forms in $\mathbb {R}^n$. In the cases $k=0$ and $k=n$, we recover well-known primal and mixed hybrid methods for the scalar Poisson equation, while for $0<k<n$, we obtain new hybrid finite element methods, including methods for the vector Poisson equation in $n=2$ and $n=3$ dimensions. We also generalize Stenberg postprocessing [RAIRO Modél. Math. Anal. Numér. 25 (1991), pp. 151–167] from $k=n$ to arbitrary $k$, proving new superconvergence estimates. Finally, we discuss how this hybridization framework may be extended to include nonconforming and hybridizable discontinuous Galerkin methods.References
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Bibliographic Information
- Gerard Awanou
- Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, 1200 W. Harrison St., Chicago, Illinois 60607
- MR Author ID: 700956
- ORCID: 0000-0001-9408-3078
- Email: awanou@uic.edu
- Maurice Fabien
- Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
- MR Author ID: 1290444
- Email: fabien@brown.edu
- Johnny Guzmán
- Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
- MR Author ID: 775211
- Email: johnny_guzman@brown.edu
- Ari Stern
- Affiliation: Department of Mathematics and Statistics, Washington University in St. Louis, One Brookings Drive, St. Louis, Missouri 06130
- MR Author ID: 880437
- Email: stern@wustl.edu
- Received by editor(s): August 11, 2020
- Received by editor(s) in revised form: October 6, 2021, and March 3, 2022
- Published electronically: September 2, 2022
- Additional Notes: This program was supported by EPSRC grant number EP/R014604/1. The first author was supported by NSF grant DMS-1720276, the third author by NSF grants DMS-1620100 and DMS-1913083, and the fourth author by NSF grant DMS-1913272.
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 79-115
- MSC (2000): Primary 65N30; Secondary 58A14
- DOI: https://doi.org/10.1090/mcom/3743
- MathSciNet review: 4496960