The Hu-Zhang element for linear elasticity on curved domains

TL;DR

This paper develops a curved domain extension of the Hu-Zhang element for linear elasticity, achieving optimal convergence rates through a novel stability analysis and local enrichment techniques.

Contribution

It introduces a new curved Hu-Zhang element with strong symmetry and H(div)-conformity, and establishes a novel inf-sup condition for stability analysis.

Findings

Optimal convergence rates for displacement and stress variables

Suboptimal stress convergence in L^2-norm without enrichment

Numerical experiments confirm theoretical predictions

Abstract

This paper extends the Hu-Zhang element for linear elasticity to curved domains, preserving strong symmetry and H(div)-conformity. The non-polynomial structure of the curved Hu-Zhang element makes it difficult to analyze the stability, which is overcome by establishing a novel inf-sup condition. Optimal convergence rates are achieved for all variables except for the stress in the $L^2$-norm. This suboptimality originates from the fact that the divergence space of the curved Hu-Zhang element is not contained in the discrete displacement space, which is improved by local $p$-enrichment on boundary elements. Two numerical experiments validate the theoretical results.