A fully-mixed finite element method for the Navier–Stokes/Darcy coupled problem with nonlinear viscosity

Abstract

We propose and analyze an augmented mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by a class of nonlinear Navier–Stokes and linear Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We apply dual-mixed formulations in both domains, and the nonlinearity involved in the Navier–Stokes region is handled by setting the strain and vorticity tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly, which yields the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. Furthermore, since the convective term in the fluid forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms arising from the constitutive and equilibrium equations of the Navier–Stokes equations, and the relation defining the strain and vorticity tensors. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with classical results on bijective monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. In particular, given an integer

A fully‐mixed finite element method for the coupling of the Navier–Stokes and Darcy–Forchheimer equations

“…According to this, we propose to look for the unknown u S in H 1 Γ S (Ω S ) and to restrict the set of corresponding test functions v S to the same space. Next, analogously to Gatica et al [14] (see also Caucao et al [16]), it is not difficult to see that the system (9) is not uniquely solvable since, given any solution ( S , S , u S , , u D , p D , ) in the indicated spaces, and given any constant c ∈ R, the vector defined by…”

Residual-Based a Posteriori Error Estimates for a Conforming Finite Element Discretization of the Navier-Stokes/Darcy Coupled Problem

“…We deduce (see [12], Subsection 5.2) the existence of a positive constant , P C independent of W and the continuous and discrete solutions, such that the following global inf-sup condition holds:…”

“…Up to the author's knowledge, the first work dealing with adaptive algorithms for the Navier-Stokes/Darcy coupling is [24], where an a posteriori error estimator for a discontinuous Galerkin approximation of this coupled problem with constant parameters is proposed. In [11], the authors have derived a reliable and efficient residual-based a posteriori error estimator for the three dimensional version of the augmented-mixed method introduced in [12]. The finite element subspaces that they have employed are piecewise constants, Raviart-Thomas elements of lowest order, continuous piecewise linear elements, and piecewise constants for the strain, Cauchy stress, velocity, and vorticity in the fluid, respectively, whereas Raviart-Thomas elements of lowest order for the velocity, piecewise constants for the pressure, and continuous piecewise linear elements for the traces, are considered in the porous medium.…”

A nonlinear Stokes–Biot model for the interaction of a non-Newtonian fluid with poroelastic media

“…In the latter case, the analysis reduces to the Hilbert space setting. Nonlinear Stokes-Darcy models with bounded viscosity have been studied in [13,20,23], while the unbounded case is considered in [22].…”