Hybridization and postprocessing in finite element exterior calculus

Awanou, Gerard; Fabien, Maurice; Guzmán, Johnny; Stern, Ari

doi:10.1090/mcom/3743

Hybridization and postprocessing in finite element exterior calculus
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by Gerard Awanou, Maurice Fabien, Johnny Guzmán and Ari Stern HTML | PDF

Math. Comp. 92 (2023), 79-115 Request permission

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.

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