### Abstract

This paper is devoted to the mathematical and numerical analysis of a model describing the interfacial 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 Darcy’s law describes fluid motion using filtration velocity and pressure. 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 concentration. 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 discretizations 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.

Original language | English |
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Number of pages | 31 |

Journal | IMA Journal of Numerical Analysis |

DOIs | |

Publication status | Accepted/In press - 13 Mar 2020 |

## Cite this

Alvarez, M., Gatica, G. N., & Ruiz Baier, R. (Accepted/In press). A mixed-primal finite element method for the coupling of Brinkman–Darcy flow and nonlinear transport.

*IMA Journal of Numerical Analysis*. https://doi.org/10.1093/imanum/drz060