A mixed formulation for the Signorini problem in nearly incompressible elasticity

We present a free-locking finite element approximation of the displacement-pressure formulation for the Signorini problem, in nearly incompressible elasticity. Studying the nonlinear variational problem requires an appropriate saddle point theory. An abstract framework is laid down and applied to the system of the variational inequalities we are involved in. Existence and uniqueness results for the continuous problem are proven and optimal convergence rates of the mixed Taylor-Hood finite element discretisation are proved. Some numerical experiences are reported to underline the reliability of this approach.