Coupling nonlinear Stokes and Darcy flow using mortar finite elements

Abstract

We study a system composed of a non-linear Stokes flow in one subdomain coupled with a nonlinear porous medium flow in another subdomain. Special attention is paid to the mathematical consequence of the shear-dependent fluid viscosity for the Stokes flow and the velocity-dependent effective viscosity for the Darcy flow. Motivated by the physical setting, we consider the case where only flow rates are specified on the inflow and outflow boundaries in both subdomains. We recast the coupled Stokes-Darcy system as a reduced matching problem on the interface using a mortar space approach. We prove a number of properties of the nonlinear interface operator associated with the reduced problem, which directly yield the existence, uniqueness and regularity of a variational solution to the system. We further propose and analyze a numerical algorithm based on mortar finite elements for the interface problem and conforming finite elements for the subdomain problems. Optimal a priori error estimates are established for the interface and subdomain problems, and a number of compatibility conditions for the finite element spaces used are discussed. Numerical simulations are presented to illustrate the algorithm and to compare two treatments of the defective boundary conditions.

Coupling Two-Phase Fluid Flow with Two-Phase Darcy Flow in Anisotropic Porous Media

“…In the past two decades, there are many previous works addressing single-phase StokesDarcy system of single-phase flows [4][5][6][7] and the related problems such as Stokes-Navier Stokes systems [8,9] and Stokes-Laplacian systems [1,10]. However research of the two-phase flow in coupled free flow and porous media regions is still at a primary stage.…”

Approximation of the Stokes–Darcy System by Optimization

“…A variety of solution algorithms have been proposed for solving the Stokes-Darcy system (see, for instance, [5] and references therein). Fully coupled approaches are considered in [1,2,21,10,24], while decoupling strategies are analyzed in [3,5,9,13,11,25]. Of the fully coupled approaches, some introduce new finite element spaces [1,2] (along with modified solution algorithms), and others either introduce Lagrange multiplier spaces [21,10] or fully discontinuous approximations [24] to resolve the coupled system.…”

“…The mortar space methods introduce a subproblem based on mass conservation on the interface [24,11,3,12], while the Robin-Robin domain decomposition methods [5,9] combine the conservation of mass and balance of normal forces on the interface into Robin conditions associated with each subproblem. A two grid solution approach was proposed and analyzed in [4], with an initial coupled approximation computed on a coarse mesh and then a correction, computable in parallel, determined on a fine mesh for the Stokes and Darcy subproblems.…”

Generalized Newtonian fluid flow through a porous medium

“…Typical assumptions are uniform continuity of ν ef f (|u|)u and strong monotonicity of ν ef f (|u|) [7,8,10], i.e., there exists C > 0 such that…”