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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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Cite

BibTeXRIS
BibTeX
@misc{hu2026sparsegrid,
  title = {{Sparse-grid discretizations of BGG sequences}},
  author = {Yihanqi Hu and Yuwen Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29516},
  eprint = {2610.29516},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29516},
}
RIS
TY  - UNPB
TI  - Sparse-grid discretizations of BGG sequences
AU  - Yihanqi Hu
AU  - Yuwen Li
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29516
UR  - https://arxiv.org/abs/2610.29516
AB  - 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.
ER  -