A fully mixed finite element method for the coupling of the Stokes and Darcy–Forchheimer problems. (1st February 2019)

• Record Type: Journal Article
• Authors: Almonacid, Javier A; Díaz, Hugo S; Gatica, Gabriel N; Márquez, Antonio
• Is Part Of: IMA journal of numerical analysis. Volume 40:Number 2(2020)
• Page Start: 1454
• Page End: 1502
• Publication Date: 2019-02-01
• Subjects: Stokes equation -- Darcy–Forchheimer equation -- fully mixed formulation -- twofold saddle point -- mixed finite element methods -- a priori error analysis; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/dry099
• Languages: English
• ISSNs: 0272-4979

Abstract

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.