A Locking-free modified conforming FEM for planar elasticity

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.