A Mixed and Nonconforming FEM with Nonmatching Meshes for a Coupled Stokes-Darcy Model

Huang, Peiqi; Chen, Jinru; Cai, Mingchao

doi:10.1007/s10915-012-9574-y

Cited by 20 publications

(7 citation statements)

References 23 publications

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“…Some preconditioning techniques and a two‐grid approach were studied in References 3,14 and 15. In References 5 and 6, a nonconforming mixed finite element method for the coupled model was presented. Most of the existing works solve the model with straight interfaces and are based on interface‐fitted meshes.…”

Section: Introductionmentioning

confidence: 99%

An extended finite element method for coupled Darcy–Stokes problems

Cao

Chen

2022

Numerical Meth Engineering

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In this article, we propose an extended finite element method to deal with coupled Darcy-Stokes problems with straight or curved interfaces. A mixed approximate form is presented based on Brezzi-Douglas-Marini finite element space and piecewise constant function space for Darcy equations, and a mixed approximate form is shown based on piecewise quadratic function space and piecewise constant function space for Stokes equations. By introducing some stabilization terms, we prove that the inf-sup condition for the discrete coupled problem is valid. Moreover, the optimal convergence is derived. Finally, some numerical examples are presented to demonstrate our theoretical results.

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Section: Introductionmentioning

confidence: 99%

An extended finite element method for coupled Darcy–Stokes problems

Cao

Chen

2022

Numerical Meth Engineering

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“…Over the last few decades, a great deal of effort has been devoted to developing an appropriate approximate solution to study the Stokes-Darcy model. Many different techniques and numerical methods were applied to investigate the Stokes-Darcy fluid flow model, such as coupled finite elements methods [15][16][17][18][19][20][21], two-grid/multi-grid methods [22][23][24][25][26][27][28], discontinuous Galerkin finite element methods [29][30][31][32], partitioned time stepping methods [33][34][35][36], least squares methods [37][38][39], domain decomposition methods [1,2,4,[40][41][42][43][44][45][46][47][48], local-parallel finite element methods [49], interface relaxation method [50,51], motar finite element methods [52,53], Lagrange multiplier methods [11,[54][55]…”

Section: Introductionmentioning

confidence: 99%

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

Mahbub

Shi

et al. 2018

Adv Differ Equ

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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.

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“…We consider the Stokes‐Darcy model for coupling fluid flow in conduits with porous media flow. This type of coupled flow is often involved in many applications, such as subsurface flow problems , industrial filtrations , and flow in vuggy porous media . The model consists of Stokes equations to govern the flow in conduits, Darcy's law to govern the flow in porous media, and three interface conditions to couple these two constituent models together.…”

Section: Introductionmentioning

confidence: 99%

A weak Galerkin finite element method for a coupled Stokes‐Darcy problem

Liu

et al. 2017

Numerical Methods Partial

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In this article, we introduce and analyze a weak Galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes equations in primal velocity-pressure formulation and Darcy equation in the second order primary formulation, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. By using the weak Galerkin approach, we consider the two-dimensional problem with the piecewise constant elements for approximations of the velocity, pressure, and hydraulic head. Stability and optimal error estimates are obtained. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the weak Galerkin approximation.Recently, the Hybridizable Discontinuous Galerkin (HDG) method has been developed based on discrete unknowns attached to mesh faces and cells for elliptic PDEs, which is originally introduced in [1] in the framework of second order elliptic problems. By exploiting Lagrange multipliers, the constraint of function continuity on the interelement boundaries is relaxed in the HDG method. We just recall the review works [1-10] and the references therein.

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A Mixed and Nonconforming FEM with Nonmatching Meshes for a Coupled Stokes-Darcy Model

Cited by 20 publications

References 23 publications

An extended finite element method for coupled Darcy–Stokes problems

An extended finite element method for coupled Darcy–Stokes problems

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

A weak Galerkin finite element method for a coupled Stokes‐Darcy problem

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