Dra. Camaño-Valenzuela, Jessika

In this article, we analyse an augmented mixed finite element method for the steady Navier–Stokes equations. More precisely, we extend the recent results from Camaño et al.. (2017, Analysis of an augmented mixed-FEM for the Navier–Stokes problem. Math. Comput., 86, 589–615) to the case of mixed no-slip and traction boundary conditions in different parts of the boundary, and introduce and analyse a new pseudostress–velocity-augmented mixed formulation for the fluid flow problem. The well-posedness analysis is carried out by combining the classical Babuška–Brezzi theory and Banach’s fixed-point theorem. 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 k≥0 and continuous piecewise polynomials of degree k+1 k+1 for the nonlinear pseudostress 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 Clément interpolant, whereas the efficiency of the estimator follows from 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.

A new mixed variational formulation for the Navier--Stokes equations with constant density and variable viscosity depending nonlinearly on the gradient of velocity, is proposed and analyzed here. Our approach employs a technique previously applied to the stationary Boussinesq problem and to the Navier--Stokes equations with constant viscosity, which consists firstly of the introduction of a modified pseudostress tensor involving the diffusive and convective terms, and the pressure. Next, by using an equivalent statement suggested by the incompressibility condition, the pressure is eliminated, and in order to handle the nonlinear viscosity, the gradient of velocity is incorporated as an auxiliary unknown. Furthermore, since the convective term forces the velocity to live in a smaller space than usual, we overcome this difficulty by augmenting the variational formulation with suitable Galerkin-type terms arising from the constitutive and equilibrium equations, the aforementioned relation defining the additional unknown, and the Dirichlet boundary condition. The resulting augmented scheme is then written equivalently as a fixed point equation, and hence the well-known Schauder and Banach theorems, combined with classical results on bijective monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. No discrete inf-sup conditions are required for the well-posedness of the Galerkin scheme, and hence arbitrary finite element subspaces of the respective continuous spaces can be utilized. In particular, given an integer K >_0, piecewise polynomials of degree _< K for the gradient of velocity, Raviart--Thomas spaces of order K for the pseudostress, and continuous piecewise polynomials of degree _< K + 1 for the velocity, constitute feasible choices. Finally, optimal a priori error estimates are derived, and several numerical results illustrating the good performance of the augmented mixed finite element method and confirming the theoretical rates of convergence are reported.

In this paper we propose and analyze 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 pseudostress 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. Further variables of interest, such as the fluid pressure, the fluid vorticity and the fluid velocity gradient, can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. 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. Then, the classical Banach fixed point theorem and the Lax-Milgram lemma are applied to prove well-posedness of the continuous problem. Similarly, we establish well-posedness and the corresponding Cea estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown. In particular, the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree K for the nonlinear-pseudostress tensor and continuous piecewise polynomial elements of degree K + 1 for the velocity, which leads to an optimal convergent scheme. In addition, we provide two iterative methods to solve the corresponding nonlinear system of equations and analyze their convergence. Finally, several numerical results illustrating the good performance of the method are provided.