A Banach spaces-based mixed finite element method for the stationary convective Brinkman–Forchheimer problem

Abstract

We propose and analyze a new mixed finite element method for the nonlinear problem given by the stationary convective Brinkman–Forchheimer equations. In addition to the original fluid variables, the pseudostress is introduced as an auxiliary unknown, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the aforementioned tensor and the velocity. As a consequence, we obtain a mixed variational formulation consisting of a nonlinear perturbation of, in turn, a perturbed saddle point problem in a Banach spaces framework. In this way, and differently from the techniques previously developed for this model, no augmentation procedure needs to be incorporated into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that recently established solvability results for perturbed saddle-point problems in Banach spaces, along with the well-known Banach–Nečas–Babuška and Banach theorems, are applied to prove the well-posedness of the continuous and discrete systems. The finite element discretization involves Raviart–Thomas elements of order $k \ge 0$ for the pseudostress tensor and discontinuous piecewise polynomial elements of degree $\le k$ for the velocity. Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method. In particular, the case of flow through a 2D porous media with fracture networks is considered.