## Preprint 2019-13

## Mario Álvarez, Gabriel N. Gatica, Ricardo Ruiz-Baier:

### A mixed-primal finite element method for the coupling of Brinkman-Darcy flow and nonlinear transport

### Abstract:

This paper is devoted to the mathematical and numerical analysis of a model describing the flow-transport interaction in a porous-fluidic domain. The medium consists of a highly permeable material, where the flow of an incompressible viscous fluid is governed by Brinkman equations (written in terms of vorticity, velocity and pressure), and a porous medium where the Darcy law describes fluid motion. Gravity and the local fluctuations of a scalar field (representing for instance, the solids volume fraction, or the concentration of a contaminant) are the main drivers of the fluid patterns on the whole domain, and the Brinkman-Darcy equations are coupled to a nonlinear transport equation accounting for mass balance of the scalar. We introduce a mixed-primal variational formulation of the problem and establish existence and uniqueness of solution using fixed-point arguments and small-data assumptions. A family of Galerkin discretisations that produce divergence-free discrete velocities is also presented and analysed using similar tools to those employed in the continuous problem. Convergence of the resulting mixed-primal finite element method is proven, and some numerical examples confirming the theoretical error bounds and illustrating the performance of the proposed discrete scheme are reported.

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

**Mario ÁLVAREZ, Gabriel N. GATICA, Ricardo RUIZ-BAIER: ***A mixed-primal finite element method for the coupling of Brinkman-Darcy flow and nonlinear transport*. IMA Journal of Numerical Analysis, vol. 41, 1, pp. 381-411, (2021).