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 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.

A Strongly Mass Conservative Method for the Coupled Brinkman-Darcy Flow and Transport

“…We assume that \Omega has a Lipschitz continuous boundary split into two disjoint subboundaries with positive measure, i.e., \partial\Omega = \Gamma B \cup \Gamma D , where \Gamma B = \partial\Omega B \setminu \Gamma and \Gamma D = \partial\Omega D \setminu \Gamma . Following [2], we adopt the following interface conditions:…”

“…In [38], a stabilized mixed finite element method in conjunction with a velocity-pressure-concentration formulation is exploited to discretize the coupled Stokes--Darcy flow and transport. In [1,2], a primal mixed finite element method in conjunction with a vorticity-velocity-pressure formulation is used for the discretization of the Brinkman--Darcy flow, and a conforming finite element method is used for the discretization of the nonlinear transport equation. In the works B167 presented in [42,12], the authors are devoted to the analysis of partitioned time stepping methods for the conforming discretizations on the two subdomains.…”

A strongly mass conservative method for the coupled Brinkman-Darcy flow and transport

“…We assume that Ω has a Lipschitz continuous boundary split into two disjoint sub-boundaries with positive measure, i.e., ∂Ω = Γ B ∪ Γ D , where Γ B = ∂Ω B \Γ and Γ D = ∂Ω D \Γ. Following [2], we adopt the following interface conditions:…”

“…In [34], a stabilized mixed finite element method in conjunction with velocity-pressure-concentration formulation is exploited to discretize the coupled Stokes-Darcy flow and transport. In [1,2], a primal mixed finite element method in conjunction with vorticity-velocity-pressure formulation is used for the discretization of the Brinkman-Darcy flow and a conforming finite element method is used for the discretization of the nonlinear transport equation. In the works presented in [38,11], the authors are devoted to the analysis of partitioned time stepping methods for the conforming discretizations on the two subdomains.…”

Assembly of multiscale linear PDE operators

“…Let next Ω i ⊂ R d , d = 2, 3, i = 1, 2 be the fluid domain and a porous domain which share a common interface Γ. A model for transport of a scalar φ in such a medium Ω = Ω 1 ∪ Ω 2 was recently analyzed by [3]. Here we shall consider a simplified, linearized version of the system…”