Error analysis of an augmented mixed method for the Navier-Stokes problem with mixed boundary conditions

In this paper we analyze an augmented mixed finite element method for the steady Navier-Stokes equations. More precisely, we extend the recent results from Camano˜ et al. (2017) to the case of mixed no-slip and traction) boundary conditions in different parts of the boundary, and introduce and analyze a new pseudostress-velocity augmented mixed formulation for the fluid flow problem. The well-posedness analysis is carried out by combining the classical Babuska-Brezzi theory and Banach’s fixed-point Theo- ˇ rem. A proper adaptation of the arguments exploited in the continuous analysis allows us to state suitable hypotheses on the finite element subspaces ensuring that the associated Galerkin scheme is well-defined. For instance, Raviart-Thomas elements of order k > 0 and continuous piecewise polynomials of degree k + 1 for the nonlinear pseudo-stress tensor and velocity, respectively, yield optimal convergence rates. In addition, we derive a reliable and efficient residual-based a posteriori error estimator for the proposed discretization. The proof of reliability hinges on the global inf–sup condition and the local approximation properties of the Clement interpolant, whereas the efficiency of the estimator follows from a inverse ´ inequalities and localization via edge–bubble functions. A set of numerical results exemplifies the performance of the augmented method with mixed boundary conditions. The tests also confirm the reliability and efficiency of the estimator, and show the performance of the associated adaptive algorithm