A fully mixed finite element method for the coupling of the Stokes and Darcy–Forchheimer problems

• Published in: IMA journal of numerical analysis Vol. 40; no. 2; pp. 1454 - 1502
• Main Authors: Almonacid, Javier A, Díaz, Hugo S, Gatica, Gabriel N, Márquez, Antonio
• Format: Journal Article
• Language: English
• Published: Oxford University Press 24-04-2020
• Subjects: twofold saddle point; fully mixed formulation; Stokes equation; error analysis; Darcy–Forchheimer equation; mixed finite element methods
• ISSN: 0272-4979; 1464-3642
• DOI: 10.1093/imanum/dry099

Abstract
In this paper we introduce and analyze a fully mixed formulation for the nonlinear problem given by the coupling of the Stokes and Darcy–Forchheimer equations with the Beavers–Joseph–Saffman condition on the interface. This new approach yields non-Hilbert normed spaces and a twofold saddle point structure for the corresponding operator equation, whose continuous and discrete solvabilities are analyzed by means of a suitable abstract theory developed for this purpose. In particular, feasible choices of finite element subspaces include PEERS of the lowest order for the stress of the fluid, Raviart–Thomas of the lowest order for the Darcy velocity, piecewise constants for the pressures and continuous piecewise linear elements for the vorticity. An a priori error estimates and associated rates of convergence are derived, and several numerical results illustrating the good performance of the method are reported.