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

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journal contribution
posted on 14.02.2020, 11:30 authored by Sergio Caucao, Marco DiscacciatiMarco Discacciati, Gabriel Gatica, Ricardo Oyarzua
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.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN)

Volume

54

Issue

5

Pages

1689 - 1723

Publisher

EDP Sciences

Version

AM (Accepted Manuscript)

Rights holder

© EDP Sciences and S.M.A.I.

Publisher statement

The original publication is available at www.esaim-m2an.org

Acceptance date

09/02/2020

Publication date

2020-07-28

Copyright date

2020

ISSN

0764-583X

eISSN

1290-3841

Language

en

Depositor

Dr Marco Discacciati. Deposit date: 13 February 2020