We propose and analyze a low-order virtual element for linear elasticity problems. It can be seen as an evolution of the Bernardi–Raugel element to general polygonal meshes. Like the Bernardi–Raugel element, the local degrees of freedoms for the new virtual element are the values of two components of $\mathbf{v}$ at vertexes and the values of the lowest moment of normal component of $\mathbf{v}$ on each edge. The proposed method is well-posed, and optimal error estimates are obtained in the $L^2$ and $H^1$ norms. Moreover, the generic constants in these estimates are shown to be uniform with respect to the Lamé coefficient $\lambda$. Numerical tests are provided to illustrate the good performance of the method and confirm our theoretical predictions.

我们提出并分析了线性弹性问题的低阶虚拟单元。可以将其视为Bernardi–Raugel元素向通用多边形网格的演变。像Bernardi–Raugel元素一样，新虚拟元素的局部自由度是以下两个分量的值：$\mathbf{v}$ 在顶点和法线分量的最低矩的值 $\mathbf{v}$在每个边缘上。提出的方法具有良好的定位条件，并给出了最优误差估计。${\mathrm{大号}}^2$ 和 $H^{\mathrm{1个}}$规范。此外，这些估计中的一般常数相对于Lamé系数显示出一致$\lambda$。提供数值测试以说明该方法的良好性能并证实我们的理论预测。