Higher-order finite element de Rham complexes on sparse grids

Yihanqi Hu; Yuwen Li

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arXiv · 2609.27768

Higher-order finite element de Rham complexes on sparse grids

Yihanqi Hu·Yuwen Li

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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.

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Yihanqi Hu, Yuwen Li. 2026-08-16. Higher-order finite element de Rham complexes on sparse grids. https://arxiv.org/abs/2609.27768

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