Stabilized finite element method for the stationary mixed Stokes–Darcy problem

Abstract

This paper considers numerical methods for solving the viscous incompressible steady-state Stokes-Darcy problem that can be implemented by the use of existing surface water and groundwater codes. In the porous medium problem for subsurface flow, a mixed discretization, which describes the macroscopic properties of a filtration process and is vigorous with respect to the variations in the material data, is often advocated. However, the theory of mixed spacial discretizations to Stokes-Darcy problems is far less developed than non-mixed versions. We develop herein a new robust stabilized fully mixed discretization technique in the porous media region coupled with the fluid region via the physically appropriate coupling conditions on the interface. The method developed here does not use any Lagrange multiplier and introduces a stabilization term in the temporal discretization to ensure the stability of the finite element scheme. The well-posedness of the finite element scheme and its convergence analysis are also derived. Finally, the efficiency and accuracy of the numerical methods are illustrated by several testing examples.

A Posteriori Error Analysis for a New Fully Mixed Isotropic Discretization of the Stationary Stokes-Darcy Coupled Problem

“…We have the following results (see [16], Theorem 2 and Theorem 3): Theorem 1. There exists a unique solution U h ∈ H h to problem (23) and if the solution U ∈ H of the continuous problem (18) is smooth enough, then we have…”

“…There are three popular formulations of the coupled Darcy-Stokes flow, namely, the primal formulation, the mixed formulation in the Darcy region, or the fully mixed formulation, see, for example, the works [8][9][10][11][12][13][14][15][16] for some mathematical analysis. Two different mixed formulations have been studied by Layton et al in [15].…”

“…In [16], a stabilized finite element method for the stationary-mixed Stokes-Darcy problem has been proposed for the fully mixed formulation. The authors have used the well-known MINI elements (P1b − P1) to approximate the velocity and pressure in the conduit for the Stokes equation.…”

“…An a priori error analysis is performed with some numerical tests confirming the convergence rates. However, to our best knowledge, they did not talk about the adaptative method for the fully mixed discretization proposed in [16]. In this case, our main objective is to perform an a posteriori error analysis by constructing reliable and efficient indicator errors.…”

Finite calculus. A stabilized finite element method for solving the Navier-Stokes equations. Application to ocean turbulence

“…We may mention the signicant work of Guasch and Codina (2009) The interest of the scientic community in stabilized methods continues to grow, judging by the number of papers citing these works. To mention some of the advances published in the past years, we nd applications in aeroelasticity (Svacek, 2004), ion ow (Chaudhry et al, 2014), ow in porous media (Yu et al, 2018), liquid membranes (Sauer, 2014), etc.…”