Luis F. Gatica

It is very well established nowadays that the possibility of computing the stress σ more accurately than the displacement u constitutes the main advantage of using dual-mixed finite element methods in elasticity. As it is well known, a dual-mixed method is characterized by the utilization of the spaces [L²(Ω)]² and H(div; Ω) to look for the u and σ, respectively. Nevertheless, this approach has not been fully investigated in the incompressible and nearly incompressible cases, for which most of the variational formulations employed are of the Stokes-like type (also named primal-mixed ones). The absence of dual-mixed formulations for incompressible elasticity has been also noticed in the works concerning a-posteriori error analysis and nonlinear constitutive equations.

In this talk we propose a new dual-mixed finite element method for nonlinear incompressible plane elasticity with mixed boundary conditions. The approach is based on the fact that the resulting variational formulation becomes a two-fold saddle point operator equation. Thus, a slight generalization of the classical Babuška-Brezzi theory is applied to show the well-posedness of the continuous and discrete formulations, and to derive the corresponding a priori error estimates. In addition, we use the Ritz projection operator to obtain a reliable and quasi-efficient a posteriori error estimate. Finally, we show several numerical results illustrating the good performance of the associated adaptive algorithm.