DG Approximation of Coupled Navier–Stokes and Darcy Equations by Beaver–Joseph–Saffman Interface Condition

Girault, Vivette; Rivière, Béatrice

doi:10.1137/070686081

Cited by 227 publications

(163 citation statements)

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“…The first equation in (4) corresponds to the balance of normal forces [19,31,39], whereas the second one is known as the Beavers-Joseph-Saffman law, which establishes that the slip velocity along Σ is proportional to the shear stress along Σ (assuming also, based on experimental evidence, that u D · t is negligible). We refer to [8,35,45] for further details on this interface condition.…”

Section: Statement Of the Model Problemmentioning

confidence: 99%

“…Up to the authors' knowledge, the first works in developing numerical methods for the Navier-Stokes/Darcy coupled problem are [31] and [6]. In [31] the authors introduce and analyze a DG discretization for the nonlinear coupled problem considering the usual nonsymmetric interior penalty Galerkin (NIPG), symmetric interior penalty Galerkin (SIPG), and incomplete interior penalty Galerkin (IIPG) bilinear forms for the discretization of the Laplacian in both media and the upwind Lesaint-Raviart discretization of the convective term in the free fluid domain. In [6] the authors extend the approach in [20] (see also [18]) and introduce an iterative subdomain method employing the velocity-pressure formulation for the Navier-Stokes equation and the primal one for the Darcy equation.…”

Section: Introductionmentioning

confidence: 99%

“…They approximate the coupled nonlinear problem using conformal finite elements in both media and study the convergence properties of Newton-like iterative methods for solving this problem. We point out that, differently from [39] and [26], the approach adopted in [31] and [6] avoids the introduction of Lagrange multipliers to impose the continuity of the normal velocity of the fluid through the interface. Indeed, this condition is imposed weakly using the primal formulation of the Darcy problem for the sole pressure unknown.…”

Section: Introductionmentioning

confidence: 99%

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

Discacciati

Oyarzúa

2016

Numer. Math.

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Citation: DISCACCIATI, M. and OYARZUA, R., 2016. A conforming mixed finite element method for the Navier Stokes/Darcy coupled problem. Numerische Mathematik, 135(2), pp. 571 606. Abstract In this paper we develop the a priori analysis of a mixed finite element method for the coupling of fluid flow with porous media flow. 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-Joseph-Saffman law. We consider the standard mixed formulation in the Navier-Stokes domain and the dual-mixed one in the Darcy region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite element subspaces defining the discrete formulation employ Bernardi-Raugel and RaviartThomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are reported.

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Section: Statement Of the Model Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Discacciati

Oyarzúa

2016

Numer. Math.

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“…A weak solution for the coupled problem is analyzed and is approximated by totally discontinuous elements in [12]. In [13] the coupling of the Navier-Stokes equation with nonhomogeneous boundary conditions is analyzed by using an implicit function theorem.…”

Section: Introductionmentioning

confidence: 99%

“…We recall that the trace operator: v → v.n is continuous from H(div, Ω) to H − 1 2 (∂Ω) and the jump (v | Ω1 −v | Ω2 ).n vanishes on Γ 12 . To make precise the sense of the operator, γ t we recall that it is the trace operator and the tangential trace operator on ∂Ω, defined by γ t (w) = w × n in dimention n = 2 and n = 3 respectively.…”

Section: Introductionmentioning

confidence: 99%

The Coupling Navier-Stokes/Darcy Systems by Vorticity-Velocity-Pressure Formulation

Younès¹,

Ouali²

2014

Global Journal of Mathematical Analysis

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In this paper We consider the Coupling Navier-Stokes/Darcy equations in a two or three dimensional domain provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We establish a coupled variational formulation of this problem with three independent unknowns: the vorticity, the velocity and the pressure. We discuss coupling conditions and we analyze the global coupled model in order to prove its well-posedness and to characterize effective algorithms to compute the solution of its numerical approximation.

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Interface control domain decomposition methods for heterogeneous problems

Discacciati

Gervasio

Quarteroni

2014

Numerical Methods in Fluids

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SUMMARYThis paper is concerned with the solution of heterogeneous problems by the interface control domain decomposition (ICDD) method, a strategy introduced for the solution of partial differential equations in computational domains partitioned into subdomains that overlap. After reformulating the original boundary value problem by introducing new additional control variables, the unknown traces of the solution at internal subdomain interfaces; the latter are determined by requiring that the (a priori) independent solutions in each subdomain undergo the minimization of a suitable cost functional.We provide an abstract formulation for coupled heterogeneous problems and a general theorem of well‐posedness for the associated ICDD problem. Then, we illustrate and validate an efficient algorithm based on the solution of the Schur‐complement system restricted solely to the interface control variables by considering two kinds of heterogeneous boundary value problems: the coupling between pure advection and advection–diffusion equations and the coupling between Stokes and Darcy equations. In the latter case, we also compare the ICDD method with a classical approach based on the Beavers–Joseph–Saffman conditions. Copyright © 2014 John Wiley & Sons, Ltd.

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DG Approximation of Coupled Navier–Stokes and Darcy Equations by Beaver–Joseph–Saffman Interface Condition

Cited by 227 publications

References 19 publications

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

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

The Coupling Navier-Stokes/Darcy Systems by Vorticity-Velocity-Pressure Formulation

Interface control domain decomposition methods for heterogeneous problems

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