Residual-Based a Posteriori Error Estimates for a Conforming Finite Element Discretization of the Navier-Stokes/Darcy Coupled Problem

Abstract

We consider in this paper, a new a posteriori residual type error estimator of a conforming mixed finite element method for the coupling of fluid flow with porous media flow on isotropic meshes. Flows are governed by the Navier-Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-JosephSaffman law. The finite element subspaces consider Bernardi-Raugel and Raviart- KOFFI WILFRID HOUEDANOU et al. 38 Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for a Lagrange multiplier defined on the interface. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. In addition, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

A Posteriori Error Estimates for a Nonconforming Finite Element Discretization of the Stokes–Biot System

“…In the latter, the volumetric deformation of the elastic porous matrix is complemented with the Darcy equation that describes the average velocity of the fluid in the pores. e model features two different kinds of coupling across the interface: Stokes-Darcy coupling [2][3][4][5][6][7][8][9][10] and fluid-structure interaction (FSI) [11][12][13][14][15]. e well-posedness of the mathematical model based on the Stokes-Biot system for the coupling between a fluid and a poroelastic structure is studied in [16][17][18][19].…”

A Posteriori Error Analysis for a Lagrange Multiplier Method for a Stokes/Biot Fluid-Poroelastic Structure Interaction Model

“…In the latter, the volumetric deformation of the elastic porous matrix is complemented with the Darcy equation that describes the average velocity of the fluid in the pores. The model features two different kinds of coupling across the interface: Stokes-Darcy coupling [2][3][4][5][6][7][8][9][10] and fluid-structure interaction (FSI) [11][12][13][14][15].…”

A New Unified Stabilized Mixed Finite Element Method of the Stokes-Darcy Coupled Problem: Isotropic Discretization

“…Since, usually, stable elements for the free fluid flow cannot been successfully applied to the porous medium flow, most of the finite element formulations developed for the Stokes-Darcy coupled problem are based on appropriate combinations of stable elements for the Stokes equations with stable elements for the Darcy equations. In [4] [6] [8]- [25], and in the references therein, we can find a large list of contributions devoted to numerically approximate the solution of this interaction problem, including conforming and nonconforming methods.…”