Uniformly $hp$-stable elements for the elasticity complex
<|reference_start|>Uniformly $hp$-stable elements for the elasticity complex: For the discretization of symmetric, divergence-conforming stress tensors in continuum mechanics, we prove inf-sup stability bounds which are uniform in polynomial degree and mesh size for the Hu--Zhang finite element in two dimensions. This is achieved via an explicit construction of a bounded right inverse of the divergence operator, with the crucial component being the construction of bounded Poincar\'e operators for the stress elasticity complex which are polynomial-preserving, in the Bernstein--Gelfand--Gelfand framework of the finite element exterior calculus. We also construct $hp$-bounded projection operators satisfying a commuting diagram property and $hp$-stable Hodge decompositions. Numerical examples are provided.<|reference_end|>
@article{aznaran2024uniformly,
    title={Uniformly $hp$-stable elements for the elasticity complex},
    author={Francis R. A. Aznaran and Kaibo Hu and Charles Parker},
    journal={arXiv preprint arXiv:2409.17414},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.17414},
    primaryClass={math.NA cs.NA}
}