A simple modification to mitigate locking in conforming FEM for nearly incompressible elasticity

Due to the divergence-instability, the accuracy of low-order conforming finite element methods (FEMs) for nearly incompressible elasticity equations deteriorates as the Lam\'e parameter $\lambda\to\infty$, or equivalently as the Poisson ratio $\nu\to1/2$. This effect is known as {\itshape locking} or {\itshape non-robustness}. For the piecewise linear case, the error in the ${\bf L}^2$-norm of the standard Galerkin conforming FEM is bounded by~$C\lambda h^2$, resulting in poor accuracy for practical values of~$h$ if $\lambda$ is sufficiently large. In this short paper, we show that the locking phenomenon can be reduced by replacing $\lambda$ with~$\lambda_h=\lambda\mu/(\mu+\lambda h/L)<\lambda$ in the stiffness matrix, where $\mu$ is the second Lam\'e parameter and $L$ is the diameter of the body $\Omega$. We prove that with this modification, the error in the ${\bf L}^2$-norm is bounded by $Ch$ for a constant $C$ that does not depend on $\lambda$. Numerical experiments confirm this convergence behaviour and show that, for practical meshes, our method is more accurate than the standard method if $\lambda$ is larger than about $\mu L/h$. Our analysis also shows that the error in the ${\bf H}^1$-norm is bounded by $C\lambda_h^{1/2}\,h$, which improves the $C\lambda^{1/2}\,h$ estimate for the case of conforming FEM.