Primal and mixed locking-free non-conforming virtual element methods for control constrained linear elasticity equations

In this paper, we present non-conforming virtual element methods (VEMs) for a primal and mixed formulation of control constrained linear elasticity equations. We follow the optimize-then-discretize scheme and use a variational discretization approach for the control variable, while the state and adjoint equations are approximated using a VEM discretization. We prove that the proposed method converges uniformly in the nearly incompressible regime and derive optimal convergence estimates in the L 2 - and H 1 -norms for the state and adjoint variables, as well as in the L 2 -norm for the optimal control. Numerical experiments are presented to confirm the theoretical results and to demonstrate the locking-free performance of the method on general polygonal meshes.