In this contribution I will present the analysis of a new augmented mixed finite element method for the Navier-Stokes problem. Our approach is based on the introduction of a 'nonlinear-pseudostress' tensor linking the stress tensor with the convective term, which leads to a mixed formulation with the nonlinear-pseudostress tensor and the velocity as the main unknowns of the system. The resulting mixed formulation is augmented by introducing
Galerkin least-squares type terms arising from the constitutive and equilibrium equations of the Navier-Stokes equations and from the Dirichlet boundary condition, which are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well-posed in the continuous and discrete level.
In the second part of the talk I will extend these results to a mixed variational formulation for the Navier-Stokes equations with variable viscosity depending nonlinearly on the gradient of velocity. Moreover I will present an a posteriori error analysis of this problem, where two different reliable and efficient residual-based a posteriori error estimators are derived.
Finally, several numerical results will be provided during the presentation to illustrate the good performance of the augmented mixed method and to confirm the a priori and a posteriori error estimates presented through the presentation.
This contribution is based on joint works with Jessika Camaño (Universidad Católica de la Santísima Concepción, Chile), Gabriel N. Gatica (Universidad de Concepción, Chile), Ricardo Oyarzúa (Universidad del Bío-Bío, Chile) and Ricardo Ruiz-Baier (University of Oxford, UK).