Robust error estimation for lowest-order approximation of nearly incompressible elasticity
Provides robust error estimators for mixed finite element methods in nearly incompressible elasticity, addressing a known bottleneck in computational mechanics.
The paper analyzes error estimation for lowest-order finite element approximations of nearly incompressible elasticity, proving robust a priori and a posteriori error bounds independent of Lamé coefficients, validated by numerical experiments.
We consider so-called Herrmann and Hydrostatic mixed formulations of classical linear elasticity and analyse the error associated with locally stabilised $P_1-P_0$ finite element approximation. First, we prove a stability estimate for the discrete problem and establish an a priori estimate for the associated energy error. Second, we consider a residual-based a posteriori error estimator as well as a local Poisson problem estimator. We establish bounds for the energy error that are independent of the Lamé coefficients and prove that the estimators are robust in the incompressible limit. A key issue to be addressed is the requirement for pressure stabilisation. Numerical results are presented that validate the theory. The software used is available online.