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A nonconforming primal hybrid finite element method for the two-dimensional vector Laplacian
Mary Barker1; Shuhao Cao2; Ari Stern3
1 Public Health Sciences Division, Fred Hutchinson Cancer Center
2 Division of Computing, Analytics, and Mathematics, University of Missouri–Kansas City
3 Department of Mathematics, Washington University in St. Louis
The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 85-106.

We introduce a nonconforming hybrid finite element method for the two-dimensional vector Laplacian, based on a primal variational principle for which conforming methods are known to be inconsistent. Consistency is ensured using penalty terms similar to those used to stabilize hybridizable discontinuous Galerkin (HDG) methods, with a carefully chosen penalty parameter due to Brenner, Li, and Sung [Math. Comp., 76 (2007), pp. 573–595]. Our method accommodates elements of arbitrarily high order and, like HDG methods, it may be implemented efficiently using static condensation. The lowest-order case recovers the P 1 -nonconforming method of Brenner, Cui, Li, and Sung [Numer. Math., 109 (2008), pp. 509–533], and we show that higher-order convergence is achieved under appropriate regularity assumptions. The analysis makes novel use of a family of weighted Sobolev spaces, due to Kondrat’ev, for domains admitting corner singularities.

Published online:
DOI: 10.5802/smai-jcm.107
Classification: 65N30, 65N15, 35Q60
Keywords: nonconforming finite element methods, hybridization, hydridizable discontinuous Galerkin methods, vector Laplacian, weighted Sobolev spaces

Mary Barker 1; Shuhao Cao 2; Ari Stern 3

1 Public Health Sciences Division, Fred Hutchinson Cancer Center
2 Division of Computing, Analytics, and Mathematics, University of Missouri–Kansas City
3 Department of Mathematics, Washington University in St. Louis
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Mary Barker; Shuhao Cao; Ari Stern. A nonconforming primal hybrid finite element method for the two-dimensional vector Laplacian. The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 85-106. doi : 10.5802/smai-jcm.107. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.107/

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