A mixed-primal finite element method for the coupling of Brinkman–Darcy flow and nonlinear transport. (2nd March 2020)

• Record Type: Journal Article
• Authors: Alvarez, Mario; Gatica, Gabriel N; Ruiz-Baier, Ricardo
• Is Part Of: IMA journal of numerical analysis. Volume 41:Number 1(2021)
• Page Start: 381
• Page End: 411
• Publication Date: 2020-03-02
• Subjects: nonlinear transport -- Brinkman–Darcy coupling -- vorticity-based formulation -- fixed-point theory -- mixed finite elements -- error analysis; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drz060
• Languages: English
• ISSNs: 0272-4979

Abstract

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.