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arXiv · 2409.17414

Uniformly $hp$-stable elements for the elasticity complex

Francis R. A. Aznaran·Kaibo Hu·Charles Parker
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Abstract

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.

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Francis R. A. Aznaran, Kaibo Hu, Charles Parker. 2024-09-25. Uniformly $hp$-stable elements for the elasticity complex. https://arxiv.org/abs/2409.17414

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