  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.
