HDG methods in finite element exterior calculus
Abstract
We develop and analyze HDG methods for two central problems in finite element exterior calculus, the Hodge-Dirac problem and the Hodge-Laplace problem, in arbitrary dimension n. Our analysis allows for equal-order polynomial spaces with 𝒪(1) penalty parameters, by contrast with previous work on the Hodge-Laplace problem requiring an underlying conforming complex and 𝒪(h) or 𝒪(h−1) penalties. For the Hodge-Dirac problem with degree-r polynomials, our error estimates give optimal order-(r+1) convergence in all form degrees under suitable regularity hypotheses. For the Hodge-Laplace problem, we prove optimal order-(r+1) convergence for the k-form solution and order-(r+1/2) convergence for the (k ± 1)-form auxiliary variables. We obtain improved order-(r+1) auxiliary-variable estimates when k=1 and/or k=n−1, which in particular sharpens some recent HDG error estimates for the two-dimensional vector Poisson equation. Our analysis also encompasses cases of lower regularity, including solutions with reentrant-corner singularities on non-convex domains, and requires only mild mesh regularity conditions. The results are illustrated by numerical experiments in dimensions two and three.
Related
- Finite element exterior calculus for spectra and pseudospectra of advection-diffusion of differential forms
- Rotating bases for finite element exterior calculus: closed-form change of basis under vertex permutations
- A finite element exterior-to-interior reconstruction in the fractional conductivity inverse problem
- Grad-div stabilized finite element Picard-Newton iterative methods for the Navier-Stokes equations
- Discrete Exterior Calculus-based Whittle-Matérn Fields on Meshes
BibTeX
@misc{reedy2026methods,
title = {{HDG methods in finite element exterior calculus}},
author = {Calvin Reedy and Ari Stern and Yunhui Xue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28995},
eprint = {2609.28995},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28995},
}
RIS
TY - UNPB
TI - HDG methods in finite element exterior calculus
AU - Calvin Reedy
AU - Ari Stern
AU - Yunhui Xue
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.28995
UR - https://arxiv.org/abs/2609.28995
AB - We develop and analyze HDG methods for two central problems in finite element exterior calculus, the Hodge-Dirac problem and the Hodge-Laplace problem, in arbitrary dimension n. Our analysis allows for equal-order polynomial spaces with 𝒪(1) penalty parameters, by contrast with previous work on the Hodge-Laplace problem requiring an underlying conforming complex and 𝒪(h) or 𝒪(h^−1) penalties. For the Hodge-Dirac problem with degree-r polynomials, our error estimates give optimal order-(r+1) convergence in all form degrees under suitable regularity hypotheses. For the Hodge-Laplace problem, we prove optimal order-(r+1) convergence for the k-form solution and order-(r+1/2) convergence for the (k ± 1)-form auxiliary variables. We obtain improved order-(r+1) auxiliary-variable estimates when k=1 and/or k=n−1, which in particular sharpens some recent HDG error estimates for the two-dimensional vector Poisson equation. Our analysis also encompasses cases of lower regularity, including solutions with reentrant-corner singularities on non-convex domains, and requires only mild mesh regularity conditions. The results are illustrated by numerical experiments in dimensions two and three.
ER -