Sparse-grid discretizations of BGG sequences

Yihanqi Hu; Yuwen Li

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arXiv · 2610.29516

Sparse-grid discretizations of BGG sequences

Yihanqi Hu·Yuwen Li

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Abstract

We develop the first tensor-product sparse-grid finite element discretizations of Bernstein--Gelfand--Gelfand (BGG) complexes. Alpert wavelets and their first and second primitives yield compatible hierarchies of discontinuous, continuous, and continuously differentiable piecewise polynomials, that serve as one-dimensional ingredients of BGG sparse-grid finite elements. We construct a bounded commuting sparse-grid projection that yields stable discrete potentials for Hessian, elasticity, and div-div complexes. Under mixed Sobolev regularity, we prove sparse-grid approximation estimates with dimension-independent convergence rates in the number of degrees of freedom. Experiments for a three-dimensional incompatibility problem and a two-dimensional biharmonic problem on an annulus illustrate the efficiency of sparse grids relative to full grids.

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Yihanqi Hu, Yuwen Li. 2026-10-21. Sparse-grid discretizations of BGG sequences. https://arxiv.org/abs/2610.29516

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