Mario Álvarez, Gabriel N. Gatica, Ricardo Ruiz-Baier:
A mixed-primal finite element method for the coupling of Brinkman-Darcy flow and nonlinear transport
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.