Due to the divergence-instability, low-order conforming finite element methods (FEMs) for nearly incompressible elasticity equations suffer from the so-called locking phenomenon as the Lam\'e parameter $\lambda\to\infty$ and consequently the material becomes more and more incompressible. For the piecewise linear case, the error in the $L^2$-norm of the standard Galerkin conforming FEM is bounded by $C_\lambda h^2$. However, $C_\lambda \to \infty$ as $\lambda \to \infty$, resulting in poor accuracy for practical values of $h$ if $\lambda$ is sufficiently large. In this short paper, we show that for 2D problems the locking phenomenon can be controlled by replacing $\lambda$ with $\lambda^\alpha$ in the stiffness matrix, for a certain choice of $\alpha=\alpha_*(h,\lambda)$ in the range $0<\alpha\le 1$. We prove that for this optimal choice of $\alpha$, the error in the $L^2$-norm is bounded by $Ch$ where $C$ does not depend on $\lambda$. Numerical experiments confirm the expected convergence behaviour and show that, for practical meshes, our locking-free method is more accurate than the standard method if the material is nearly incompressible. Our analysis also shows that the error in the $H^1$-norm is bounded by $Ch^{1/2}$, but our numerical experiments suggest that this bound is not sharp.

翻译：由于散度不稳定性，低阶协调有限元法在求解近不可压缩弹性方程时，当拉梅参数$\lambda\to\infty$（即材料趋于完全不可压缩）时，会遭受所谓的闭锁现象。对于分段线性情况，标准伽辽金协调有限元法在$L^2$范数下的误差界为$C_\lambda h^2$。然而，当$\lambda \to \infty$时$C_\lambda \to \infty$，导致若$\lambda$足够大，在实际$h$值下精度严重下降。在这篇短文中，我们证明对于二维问题，通过在刚度矩阵中将$\lambda$替换为$\lambda^\alpha$，并选取特定范围内（$0<\alpha\le 1$）的$\alpha=\alpha_*(h,\lambda)$，可以控制闭锁现象。我们证明，对于这一最优的$\alpha$选择，$L^2$范数误差以$Ch$为界，其中$C$不依赖于$\lambda$。数值实验验证了预期的收敛行为，并表明对于实际网格，当材料近不可压缩时，我们的无闭锁方法比标准方法更精确。我们的分析还表明$H^1$范数误差以$Ch^{1/2}$为界，但数值实验暗示该界限并非尖锐。