Uniformly $hp$-stable elements for the elasticity complex
Numerical Analysis
2024-09-27 v1 Numerical Analysis
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 -bounded projection operators satisfying a commuting diagram property and -stable Hodge decompositions. Numerical examples are provided.
Cite
@article{arxiv.2409.17414,
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}
}