Dra. Camaño-Valenzuela, Jessika

In this paper, we propose and analyze a new momentum conservative mixed finite element method for the Navier–Stokes problem posed in nonstandard Banach spaces. Our approach is based on the introduction of a pseudostress tensor relating the velocity gradient with the convective term, leading to a mixed formulation where the aforementioned pseudostress tensor and the velocity are the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree K for the pseudostress tensor and discontinuous piece–wise polynomial elements of degree K for the velocity. With this choice of spaces, the equilibrium equation is exactly satisfied if the external force belongs to the velocity discrete space, thus the method conserves momentum, which constitutes one of the main feature of our approach. For both, the continuous and discrete problems, the Banach–Nečas–Babuška and Banach's fixed-point theorems are employed to prove unique solvability. We also provide the convergence analysis and particularly prove that the error decay with optimal rate of convergence. 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. Finally, several numerical results illustrating the performance of the method are provided.

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.

In this paper we develop an a posteriori error analysis of a new momentum conservative mixed finite element method recently introduced for the steady-state Navier–Stokes problem in two and three dimensions. More precisely, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such as local estimates, and suitable Helmholtz decompositions, we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed finite element scheme on arbitrary (convex or non-convex) polygonal and polyhedral regions. On the other hand, inverse inequalities, the localization technique based on bubble functions, among other tools, are employed to prove the efficiency of the proposed a posteriori error indicator. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.