A conforming mixed finite element method for the Navier-Stokes/Darcy-Forchheimer coupled problem
CaucaoSergio
DiscacciatiMarco
GaticaGabriel
OyarzuaRicardo
2020
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.