# A low-order nonconforming method for linear elasticity on general meshes

## TL;DR

This paper introduces a low-order, nonconforming finite element method for linear elasticity that works on general meshes, ensuring stability and convergence without locking, supported by theoretical proofs and numerical validation.

## Contribution

It develops a novel low-order nonconforming method for elasticity on arbitrary meshes, with a new stabilization term and proof of discrete Korn inequality for stability.

## Key findings

- Locking-free error estimates in energy and L2 norms.
- Convergence rates of h and h^2 for smooth solutions.
- Numerical validation confirms theoretical results.

## Abstract

In this work we construct a low-order nonconforming approximation method for linear elasticity problems supporting general meshes and valid in two and three space dimensions. The method is obtained by hacking the Hybrid High-Order method, that requires the use of polynomials of degree $k\ge1$ for stability. Specifically, we show that coercivity can be recovered for $k=0$ by introducing a novel term that penalises the jumps of the displacement reconstruction across mesh faces. This term plays a key role in the fulfillment of a discrete Korn inequality on broken polynomial spaces, for which a novel proof valid for general polyhedral meshes is provided. Locking-free error estimates are derived for both the energy- and the $L^2$-norms of the error, that are shown to convergence, for smooth solutions, as $h$ and $h^2$, respectively (here, $h$ denotes the meshsize). A thorough numerical validation on a complete panel of two- and three-dimensional test cases is provided.