A conforming mixed finite element method for the Navier–Stokes/Darcy–Forchheimer coupled problem

Abstract

In this work we present and analyse a mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by the Navier--Stokes and the Darcy--Forchheimer equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers--Joseph--Saffman law. We consider the standard mixed formulation in the Navier--Stokes domain and the dual-mixed one in the Darcy--Forchheimer region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The well-posedness of the problem is achieved by combining a fixed-point strategy, classical results on nonlinear monotone operators and the well-known Schauder and Banach fixed-point theorems. As for the associated Galerkin scheme we employ Bernardi--Raugel and Raviart--Thomas elements for the velocities, and piecewise constant elements for the pressures and the Lagrange multiplier, whereas its existence and uniqueness of solution is established similarly to its continuous counterpart, using in this case the Brouwer and Banach fixed-point theorems, respectively. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, we report some numerical examples confirming the predicted rates of convergence, and illustrating the performance of the method.

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

“…We begin by noting that hypothesis (A 1 ) implies that the nonlinear operator a is continuous, and hence obviously hemi-continuous. In this way, as a consequence of hypotheses (A 1 ) − (A 2 ) we deduce the well posedness of the problem (27) (see Caucao et al [9] [Theorem 3.1] for details). In turn, given 1 , 2 ∈Ỹ for which t 0 ( 1 ) and t 0 ( 2 ) satisfy (27), we deduce that […”

“…According to the above bibliographic discussion, the purpose of the present paper is to extend the results obtained in References [9, 12] to the coupling of the Navier–Stokes and Darcy–Forchheimer problems with constant density and viscosity, but unlike [9], by considering now dual‐mixed formulations in both domains. We introduce the pseudostress tensor as in Camaño et al [13] and subsequently eliminate the pressure unknown using the incompressibility condition.…”

“…Then, similarly to References [14,18] and [9], we test the first equations of (5) and (6) with arbitrary S ∈ H(div; Ω S ) and v D ∈ H 3 Γ D (div; Ω D ), respectively, integrate by parts, utilize the fact that…”

“…Notice that H 3 (div; Ω D ) = H (div; Ω D ) ∩ L 3 (Ω D ), which guarantees that v D ⋅ n is well defined for (see Caucao et al [9] [section 2.2] for details). In addition, analogously to Gatica et al [14] (see also Caucao et al [16]) we need to introduce the space of traces , where …”

“…Equivalently, this shows that given with ψ 1 ≠ ψ 2 the solutions t 0 ( ψ 1 ) and t 0 ( ψ 2 ) of (27) are in fact different. Now, in order to obtain (28), we proceed similarly to Scheurer [10] [Proposition 2.3] (see also Caucao et al [9] [theorem 3.1]). In fact, given , we take in (27), and observe that Then, combining hypotheses (A 1 ) − (A 2 ), estimate (22), and simple algebraic manipulation, yields (28).…”