Boundary integral solutions of coupled Stokes and Darcy flows

Tlupova, Svetlana; Cortez, Ricardo

doi:10.1016/j.jcp.2008.09.011

Cited by 70 publications

(61 citation statements)

References 39 publications

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“…Several methods have been developed to numerically solve the Stokes-Darcy problem, including coupled finite element methods [2,8,10,25,32], domain decomposition methods [12][13][14][15][16][17][18]23], Lagrange multiplier methods [20,26], two grid methods [29], discontinuous Galerkin methods [19,31], and boundary integral methods [35]. Many other methods have been developed to solve the Stokes-Brinkman and other similar models; see [1,3,[5][6][7]28,30,34,36,37] and the reference cited therein.…”

mentioning

confidence: 99%

A Parallel Robin–Robin Domain Decomposition Method for the Stokes–Darcy System

Chen¹,

Gunzburger²,

Hua³

et al. 2011

SIAM J. Numer. Anal.

148

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Domain decomposition methods for solving the coupled Stokes-Darcy system with the Beavers-Joseph interface condition are proposed and analyzed. Robin boundary conditions are used to decouple the Stokes and Darcy parts of the system. Then, parallel and serial domain decomposition methods are constructed based on the two decoupled sub-problems. Convergence of the two methods is demonstrated and the results of computational experiments are presented to illustrate the convergence.

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mentioning

confidence: 99%

A Parallel Robin–Robin Domain Decomposition Method for the Stokes–Darcy System

Chen¹,

Gunzburger²,

Hua³

et al. 2011

SIAM J. Numer. Anal.

148

View full text Add to dashboard Cite

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“…In order to compute R, we need a grid on a mesh that is highly refined close to the singular boundary points. But once R is obtained we shall solve (22) and (23) on a coarse grid. This section defines R. Section 6 is about the fast construction of R. Section 7 is on the reconstruction of ρ(z) on the fine grid from computed values on the coarse grid.…”

Section: Compressed Discretizationmentioning

confidence: 99%

“…A mesh with n such subdivision is called an n-ply refined mesh. Let K denote the integral operator in (22) and (23). Using Nyström discretization we arrive at two grids and two linear systems…”

Section: Compressed Discretizationmentioning

confidence: 99%

“…More generally, elliptic problems for multiphase materials where some continuity conditions hold on internal interfaces and loads are applied to a connected outer boundary belong to this class. Grain boundary diffusion in finite-size polycrystals [20,24] and coupled Stokes and Darcy flow [22] are two examples. Solvers based on integral equations, which are superior for pure boundary conditions, are not always applicable for mixed conditions.…”

Section: Introductionmentioning

confidence: 99%

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Integral equation methods for elliptic problems with boundary conditions of mixed type

Helsing

2009

Journal of Computational Physics

135

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Laplace's equation with mixed boundary conditions, that is, Dirichlet conditions on parts of the boundary and Neumann conditions on the remaining contiguous parts, is solved on an interior planar domain using an integral equation method. Rapid execution and high accuracy is obtained by combining equations which are of Fredholm's second kind with compact operators on almost the entire boundary with a recursive compressed inverse preconditioning technique. Then an elastic problem with mixed boundary conditions is formulated and solved in an analogous manner and with similar results. This opens up for the rapid and accurate solution of several elliptic problems of mixed type.

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“…This case of a porous medium surrounded by a Stokes flow has been a topic of active investigation with the main goal of deriving appropriate boundary conditions at the interface (Beavers & Joseph 1967;Beavers, Sparrow & Magnuson 1970;Richardson 1971;Saffman 1971;Taylor 1971;Neale & Nader 1974;Howes & Whitaker 1985;Goyeau et al 2003;Chandesris & Jamet 2006;Valdés-Parada, Goyeau & Ochoa-Tapia 2007;Tlupova & Cortez 2009;Valdés-Parada et al 2009, 2013.…”

Section: Introductionmentioning

confidence: 99%

Boundary integral formulation for flows containing an interface between two porous media

2017

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A system of boundary integral equations is derived for flows in domains composed of a porous medium of permeability k 1 , surrounded by another porous medium of different permeability, k 2 . The incompressible Brinkman equation is used to describe the flow in the porous media. We first apply a boundary integral representation of the Brinkman flow on each side of the dividing interface, and impose continuity of the velocity at the interface to derive the final formulation in terms of the interfacial velocity and surface forces. We discuss relations between the surface stresses based on the additional conditions imposed at the interface that depend on the porosity and permeability of the media and the structural composition of the interface. We present simulated results for test problems and different interface stress conditions. The results show significant sensitivity to the choice of the interface conditions, especially when the permeability is large. Since the Brinkman equation approaches the Stokes equation when the permeability approaches infinity, our boundary integral formulation can also be used to model the flow in sub-categories of Stokes-Stokes and Stokes-Brinkman configurations by considering infinite permeability in the Stokes fluid domain.

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Boundary integral solutions of coupled Stokes and Darcy flows

Cited by 70 publications

References 39 publications

A Parallel Robin–Robin Domain Decomposition Method for the Stokes–Darcy System

A Parallel Robin–Robin Domain Decomposition Method for the Stokes–Darcy System

Integral equation methods for elliptic problems with boundary conditions of mixed type

Boundary integral formulation for flows containing an interface between two porous media

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