## Preprint 2016-27

## Luis F. Gatica, Ricardo Oyarzúa, Nestor Sánchez:

### A priori and a posteriori error analysis of an augmented mixed-FEM for the Navier-Stokes-Brinkman problem

### Abstract:

We introduce and analyze an augmented mixed finite element method for the Navier-Stokes- Brinkman problem. We employ a technique previously applied to the stationary Navier- Stokes equation, which consists of the introduction of a modified pseudostress tensor relating the gradient of the velocity and the pressure with the convective term, and propose a pseudostress-velocity formulation for the model problem. Since the convective term forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Banach fixed point theorem, combined with the Lax-Milgram lemma, are applied to prove the unique solvability of the continuous and discrete systems. We point out that no discrete inf-sup conditions are required for the solvability analysis, and hence, in particular for the Galerkin scheme, arbitrary finite element subspaces of the respective continuous spaces can be utilized. For instance, given an integer k \ge 0, the Raviart-Thomas spaces of order k and continuous piecewise polynomials of degree \le k + 1 constitute feasible choices of discrete spaces for the pseudostress and the velocity, respectively, yielding optimal convergence. In addition, we derive a reliable and efficient residual-based a posteriori error estimator for the augmented mixed method. The proof of reliability makes use of a global inf-sup condition, a Helmholtz decomposition, and local approximation properties of the Clement interpolant and Raviart-Thomas operator. On the other hand, inverse inequalities, the localization technique based on element-bubble and edge-bubble functions, approximation properties of the L2-orthogonal projector, and known results from previous works, are the main tools for proving the efficiency of the estimator. Finally, some numerical results illustrating the performance of the augmented mixed method, confirming the theoretical rate of convergence and properties of the estimator, and showing the behaviour of the associated adaptive algorithms, are reported.

This preprint gave rise to the following definitive publication(s):

**Luis F. GATICA, Ricardo OYARZúA, Nestor SáNCHEZ: ***A priori and a posteriori error analysis of an augmented mixed-FEM for the Navier-Stokes-Brinkman problem*. Computers & Mathematics with Applications, vol. 75, 7, pp. 2420-2444, (2018).