In this paper, we investigate a low-order robust numerical method for the linear elasticity problem. The method is based on a Bernardi–Raugel-like \(\varvec{H}(\textrm{div})\)-conforming method proposed first for the Stokes flows in [Li and Rui, IMA J. Numer. Anal. 42 (2022) 3711–3734]. Therein, the lowest-order \(\varvec{H}(\textrm{div})\)-conforming Raviart–Thomas space (\(\varvec{RT}_0\)) was added to the classical conforming \(\varvec{P}_1\times P_0\) pair to meet the inf-sup condition, while preserving the divergence constraint and some important features of conforming methods. Due to the inf-sup stability of the \(\varvec{P}_1\oplus \varvec{RT}_0\times P_0\) pair, a locking-free elasticity discretization with respect to the Lamé constant \(\lambda \) can be naturally obtained. Moreover, our scheme is gradient-robust for the pure and homogeneous displacement boundary problem, that is, the discrete \(\varvec{H}^1\)-norm of the displacement is \(\mathcal {O}(\lambda ^{-1})\) when the external body force is a gradient field. We also consider the mixed displacement and stress boundary problem, whose \(\varvec{P}_1\oplus \varvec{RT}_0\) discretization should be carefully designed due to a consistency error arising from the \(\varvec{RT}_0\) part. We propose both symmetric and nonsymmetric schemes to approximate the mixed boundary case. The optimal error estimates are derived for the energy norm and/or \(\varvec{L}^2\)-norm. Numerical experiments demonstrate the accuracy and robustness of our schemes.

在本文中，我们研究了线性弹性问题的低阶鲁棒数值方法。该方法基于类似 Bernardi-Raugel 的\(\varvec{H}(\textrm{div})\)一致方法，首先在 [Li and Rui, IMA J. Numer.肛门。42（2022）3711–3734]。其中，最低阶\(\varvec{H}(\textrm{div})\)符合 Raviart–Thomas 空间 ( \(\varvec{RT}_0\) ) 被添加到经典符合\(\varvec {P}_1\times P_0\)对满足 inf-sup 条件，同时保留发散约束和一致方法的一些重要特征。由于\(\varvec{P}_1\oplus \varvec{RT}_0\times P_0\)对的 inf-sup 稳定性，相对于 Lamé 常数\(\lambda \) 的无锁定弹性离散化自然可以得到。此外，我们的方案对于纯齐次位移边界问题是梯度鲁棒的，即位移的离散\(\varvec{H}^1\) -范数为\(\mathcal {O}(\lambda ^ {-1})\)当外部体力是梯度场时。我们还考虑了混合位移和应力边界问题，由于\ (\varvec{RT} _0\)部分。我们提出对称和非对称方案来近似混合边界情况。最佳误差估计是针对能量范数和/或\(\varvec{L}^2\) -范数导出的。数值实验证明了我们方案的准确性和鲁棒性。