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 Lamé parameter λ → ∞ , or equivalently as the Poisson ratio ν → 1 / 2 . This effect is known as locking or non-robustness . For the piecewise linear case, the error in the L 2 ( Ω ) -norm of the standard Galerkin conforming FEM is bounded by C λ h 2 , resulting in poor accuracy for practical values of h if λ is sufficiently large. In this paper, we show that the locking phenomenon can be reduced by replacing λ with λ h = λ μ / μ 2 + ( λ h / L ) 2 < λ or λ h = λ μ / ( μ + λ h / L ) < λ in the stiffness matrix, where μ is the second Lamé parameter and L is the diameter of the body Ω . We prove that with this modification, the error in the L 2 ( Ω ) -norm is bounded by C h for a constant C that does not depend on λ . Numerical experiments confirm this convergence behaviour and show that, for practical meshes, our method is more accurate than the standard method if λ is larger than about μ L / h . Our analysis also shows that the error in the H 1 ( Ω ) -norm is bounded by C λ h 1 / 2 h , which improves the C λ 1 / 2 h estimate for the case of conforming FEM.