Hybridization and postprocessing in finite element exterior calculus

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 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.

arXiv comment30 pages; v2: major revisions to reduce length (cut review material from Section 2, eliminated Sections 3.4-3.5, fewer numerical experiments in Section 7), new results on trace norms (Lemma 2.3) and well-posedness (Theorem 3.4), and minor edits

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Cited by in corpus (1)

• An Extended Galerkin analysis in finite element exterior calculus