Higher-order finite element de Rham complexes on sparse grids
Abstract
We construct, for the first time, a family of higher-order finite element differential forms on tensor-product sparse grids. The construction starts from compatible one-dimensional spaces of continuous piecewise polynomials and discontinuous piecewise polynomials of one degree lower, linked by differentiation in each coordinate direction. Their hierarchical decompositions combine Alpert multiwavelets with their integrated counterparts. We establish commuting canonical interpolation operators and corresponding approximation error bounds under mixed Sobolev regularity. For the sparse-grid de Rham complex of arbitrary polynomial degree on the unit cube in arbitrary dimension, we also prove exactness, polynomial-degree-robust stable discrete potentials, and an H(d)-bounded commuting projection by developing a novel stable homotopy operator. Numerical experiments for curl-curl source problems on a cube and a Maxwell eigenproblem on a square-annulus illustrate the effectiveness of the proposed higher-order sparse-grid method.
Related
- Discrete Poincaré and Bogovskii operators for higher order finite element de Rham complexes
- On Finite Groups with Element Order Set of Width 3
- High-order energy diminishing ALE-SAV finite element methods for two-phase Navier--Stokes flow
- Arbitrarily High-Order Structure-Preserving Parametric Finite Element Methods for Geometric PDEs via Local Pullback Flow Maps
- One-Sided p-Explicit Stability for High-Order Cut Finite Element Nitsche Methods
BibTeX
@misc{hu2026higherorder,
title = {{Higher-order finite element de Rham complexes on sparse grids}},
author = {Yihanqi Hu and Yuwen Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27768},
eprint = {2609.27768},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27768},
}
RIS
TY - UNPB
TI - Higher-order finite element de Rham complexes on sparse grids
AU - Yihanqi Hu
AU - Yuwen Li
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27768
UR - https://arxiv.org/abs/2609.27768
AB - We construct, for the first time, a family of higher-order finite element differential forms on tensor-product sparse grids. The construction starts from compatible one-dimensional spaces of continuous piecewise polynomials and discontinuous piecewise polynomials of one degree lower, linked by differentiation in each coordinate direction. Their hierarchical decompositions combine Alpert multiwavelets with their integrated counterparts. We establish commuting canonical interpolation operators and corresponding approximation error bounds under mixed Sobolev regularity. For the sparse-grid de Rham complex of arbitrary polynomial degree on the unit cube in arbitrary dimension, we also prove exactness, polynomial-degree-robust stable discrete potentials, and an H(d)-bounded commuting projection by developing a novel stable homotopy operator. Numerical experiments for curl-curl source problems on a cube and a Maxwell eigenproblem on a square-annulus illustrate the effectiveness of the proposed higher-order sparse-grid method.
ER -