Uniformly $hp$-stable elements for the elasticity complex

TL;DR

This paper establishes uniform stability bounds for Hu--Zhang finite elements in elasticity, using explicit divergence inverses and polynomial-preserving operators within the finite element exterior calculus framework.

Contribution

It introduces an explicit construction of bounded right inverses and polynomial-preserving Poincaré operators for the elasticity complex, ensuring $hp$-stability in finite element discretizations.

Findings

Proved inf-sup stability bounds uniform in polynomial degree and mesh size.

Constructed $hp$-bounded projection operators with commuting diagram properties.

Provided numerical examples demonstrating theoretical results.

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.