Coupled Stokes Darcy Problem Research Articles

A unified local projection-based stabilized virtual element method for the coupled Stokes-Darcy problem

A unified local projection-based stabilized virtual element method for the coupled Stokes-Darcy problem

A mortar method for the coupled Stokes-Darcy problem using the MAC scheme for Stokes and mixed finite elements for Darcy

A mortar method for the coupled Stokes-Darcy problem using the MAC scheme for Stokes and mixed finite elements for Darcy

A pressure-robust mixed finite element method for the coupled Stokes–Darcy problem

A pressure-robust numerical method for the coupled Stokes–Darcy problem is proposed. We use the continuous piecewise linear polynomial space combined with the lowest order Raviart–Thomas space to discretize the Stokes equation in the fluid region and the lowest order Raviart–Thomas elements to discretize the Darcy equation in the porous media domain. We also perform a divergence-free reconstruction of the test function for the right-hand term. Our method has good properties of convergence, pressure-robustness and local mass conservation in the sense of projection. A discrete inf-sup condition and error estimates for the numerical scheme are demonstrated, the error of velocity is independent of viscosity and pressure. Finally, the theoretical analysis is validated with several examples, demonstrating the superiority of the pressure-robust scheme.

A Robin-Type Domain Decomposition Method for a Novel Mixed-Type DG Method for the Coupled Stokes--Darcy Problem

In this paper, we first propose and analyze a novel mixed-type DG method for the coupled Stokes-Darcy problem on simplicial meshes. The proposed formulation is locally conservative. A mixed-type DG method in conjunction with the stress-velocity formulation is employed for the Stokes equations, where the symmetry of stress is strongly imposed. The staggered DG method is exploited to discretize the Darcy equations. As such, the discrete formulation can be easily adapted to account for the Beavers-Joseph-Saffman interface conditions without introducing additional variables. Importantly, the continuity of normal velocity is satisfied exactly at the discrete level. A rigorous convergence analysis is performed for all the variables. Then we devise and analyze a domain decomposition method via the use of Robin-type interface boundary conditions, which allows us to solve the Stokes subproblem and the Darcy subproblem sequentially with low computational costs. The convergence of the proposed iterative method is analyzed rigorously. In particular, the proposed iterative method also works for very small viscosity coefficient. Finally, several numerical experiments are carried out to demonstrate the capabilities and accuracy of the novel mixed-type scheme, and the convergence of the domain decomposition method.

A second order multirate scheme for the evolutionary Stokes–Darcy model

A second order multirate scheme for the evolutionary coupled Stokes–Darcy problem is proposed and analyzed. The scheme decouples the problem into two smaller subproblems at each time step by exploiting the mismatch in flow activity between the domains. Specifically, the slower Darcy subproblem is integrated with a larger time step compared to the more active Stokes subproblem. This avoids expending unnecessary computational effort on sampling the slow changing Darcy solution. The slow varying Darcy solution is extrapolated/interpolated onto the fine grid for the integration of the Stokes subproblem. The long-time stability and convergence of the method is proved under a time step restriction that depends on physical parameters and the ratio of the time steps in each subdomain. The scheme includes a stabilization term that relaxes the time step restriction. Numerical experiments confirm convergence of the method and the effectiveness of the stabilization technique at relaxing the stability time step restriction.

Modified formulation of the interfacial boundary condition for the coupled Stokes–Darcy problem

Modified formulation of the interfacial boundary condition for the coupled Stokes–Darcy problem

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

In this paper, we develop an a-priori error analysis of a new unified mixed finite element method for the coupling of fluid flow with porous media flow in RN, N ∈ {2,3}, on isotropic meshes. Flows are governed by the 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. The approach utilizes a modification of the Darcy problem which allows us to apply a variant nonconforming Crouzeix-Raviart finite element to the whole coupled Stokes-Darcy problem. The well-posedness of the finite element scheme and its convergence analysis are derived. Finally, the numerical experiments are presented, which confirm the excellent stability and accuracy of our method.

A lowest-order staggered DG method for the coupled Stokes–Darcy problem

Abstract In this paper we propose a locally conservative, lowest-order staggered discontinuous Galerkin method for the coupled Stokes–Darcy problem on general quadrilateral and polygonal meshes. This model is composed of Stokes flow in the fluid region and Darcy flow in the porous media region, coupling together through mass conservation, balance of normal forces and the Beavers–Joseph–Saffman condition. Stability of the proposed method is proved. A new regularization operator is constructed to show the discrete trace inequality. Optimal convergence estimates for all the approximations covering low regularity are achieved. Numerical experiments are given to illustrate the performances of the proposed method. The numerical results indicate that the proposed method can be flexibly applied to rough grids such as the trapezoidal grid and $h$-perturbation grid.

A hybrid high-order method for a coupled Stokes-Darcy problem on general meshes

In this work, a hybrid high-order (HHO) method on general meshes is presented to solve a coupled Stokes-Darcy problem with the Beavers-Joseph-Saffman interface condition. Constructed on polynomials of arbitrary degree k≥0, the numerical method is established in terms of discrete unknowns attached to mesh faces and cells (or elements). The unified discrete scheme for Stokes equation and Darcy equation is given by the continuity condition of the interface. The unique solvability of the discrete scheme is proved. Moreover, the energy error estimate for the velocity and L2-error estimate for pressure of order (k+1) are derived. Finally, a series of numerical experiments are reported to illustrate the accuracy, mass conservation and robustness of our method.

A virtual element method for the coupled Stokes–Darcy problem with the Beaver–Joseph–Saffman interface condition

In this work, we propose a virtual element method for discretizing the equations that couple the incompressible steady Stokes flow with the Darcy flow by means of the Beaver–Joseph–Saffman condition on their interface. In addition to avoiding explicit expressions of basis functions, this method can not only improve the computational efficiency of any polynomial degree, but also can treat any polygonal elements, including non-convex and non-matching elements. Moreover, combining with the discrete inf-sup condition of a virtual element approximation for the velocity and pressure pair $$P_{k}/P_{k-1}$$, we can obtain optimal error estimates. Furthermore, numerical experiments are presented to show the efficiency and validity of the coupled method.

A Stabilized Finite Volume Element Method for Stationary Stokes–Darcy Equations Using the Lowest Order

We present a stabilized finite volume element method for the coupled Stokes–Darcy problem with the lowest order [Formula: see text] element for the Stokes region and [Formula: see text] element for the Darcy region. Based on adding a jump term of discrete pressure to the approximation equation, a discrete inf-sup condition is established for the proposed method. The optimal error estimates in the [Formula: see text]-norm for the velocity and piezometric head and in the [Formula: see text]-norm for the pressure are proved. And they are also verified through some numerical experiments. Two figures are given to show the full comparison for the local mass conservation between the proposed method and the stabilized finite element method. And this method can also be computed directly in the irregular domain according to the last experiment.

A Discrete Duality Finite Volume Method for Coupling Darcy and Stokes Equations

Abstract We present a general finite volume method to solve a coupled Stokes-Darcy problem, we propose two domains corresponding to fluid region and porous region with a physical intersection. At the contact interface between the fluid region and the porous media we impose two conditions; the first one is the normal continuity of the velocity and the second one is the continuity of the pressure. Furthermore, due to the lack of information about both the velocity and the pressure on the interface, we will use Schwarz domain decomposition. In Darcy equations, the tensor of permeability will be considered as variable, since it depends on both the properties of the porous medium and the viscosity of the fluid. Numerical examples are presented to demonstrate the efficiency of the proposed method.

Finite volume method for a coupled Stokes-Darcy problem


 
 
 In this paper we propose a finite volume method to solve the coupled Stokes-Darcy problem using steady Stokes equations for the fluid region and Darcy equations for the porous region.
 At the contact interface between the fluid region and the porous media we imposed two conditions. The first one is the normal continuity of the velocity, while the second one is the continuity of the pressure. Furthermore, due to the lack of information about both the velocity and the pressure on the interface, we will use schwarz domain decomposition method. In Darcy equations, the tensor of permeability will be considered as variable, since it depends on both the prop- erties of the porous medium and the viscosity of the fluid. Numerical examples are presented to demonstrate the efficiency of the proposed method
 
 

A unified mixed finite element approximations of the Stokes–Darcy coupled problem

In this paper we develop and analyze a unified approximation of the velocity–pressure pair for the Stokes–Darcy coupled problem in a plane domain. It is well known that, stable finite element approximations for the Stokes problem may not be appropriate for Darcy problem and for the coupling of fluid flow (modeled by the Stokes equations) with porous media flow (modeled by the Darcy equation), and therefore, different spaces are commonly used for the discretizations of the Darcy and the Stokes problems. In this work we proposed a modification of the Darcy problem which allows us to apply the classical Mini-element to the whole coupled Stokes–Darcy problem. The proposed method is probably one of the cheapest method for continuous approximation of the coupled system, has optimal accuracy with respect to solution regularity, and has simple and straightforward implementations. Numerical experiments are also presented, which confirm the excellent stability and accuracy of our method.

A multi-grid technique for coupling fluid flow with porous media flow

In this paper, we consider the coupling of fluid flow with porous media flow. A multi-grid finite element method for the coupled Stokes–Darcy problem with the Beavers–Joseph interface condition is proposed and discussed. The optimal error estimates are obtained. Numerical experiment is given to verify the theoretical analysis and indicate the accuracy and efficiency of the multi-grid method.

Two-grid finite element method for the stabilization of mixed Stokes-Darcy model

A two-grid discretization for the stabilized finite element method for mixed Stokes-Darcy problem is proposed and analyzed. The lowest equal-order velocity-pressure pairs are used due to their simplicity and attractive computational properties, such as much simpler data structures and less computer memory for meshes and algebraic system, easier interpolations, and convenient usages of many existing preconditioners and fast solvers in simulations, which make these pairs a much popular choice in engineering practice; see e.g., [4,27]. The decoupling methods are adopted for solving coupled systems based on the significant features that decoupling methods can allow us to solve the submodel problems independently by using most appropriate numerical techniques and preconditioners, and also to reduce substantial coding tasks. The main idea in this paper is that, on the coarse grid, we solve a stabilized finite element scheme for coupled Stokes-Darcy problem; then on the fine grid, we apply the coarse grid approximation to the interface conditions, and solve two independent subproblems: one is the stabilized finite element method for Stokes subproblem, and another one is the Darcy subproblem. Optimal error estimates are derived, and several numerical experiments are carried out to demonstrate the accuracy and efficiency of the two-grid stabilized finite element algorithm.

A weak Galerkin finite element method for a coupled Stokes–Darcy problem on general meshes

In this paper, 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 usual polynomials of degree k ≥ 1 for the velocity and hydraulic head, while polynomials of degree k − 1 for the pressure, the velocity and hydraulic head is enhanced by polynomials of degree k − 1 on the edge of a finite element partition. This new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained. Moreover, numerical experiences are presented to illustrate the good performance, confirm the optimal order of convergence and verify the efficiency of the proposed weak Galerkin method in this paper. We also deal with finite element partitions consisting of general meshes, such as triangular mesh, quadrilateral mesh, hexagonal-dominant mesh and Voronoi mesh for the numerical weak Galerkin approximation.

A stabilized finite volume element method for a coupled Stokes–Darcy problem

In this paper, we present a stabilized finite volume element method with the conforming finite element triples P1–P0–P1 and P1–P1–P1 for approximating the velocity, pressure, and hydraulic head of a coupled Stokes–Darcy problem. The proposed method is convenient to implement, computationally efficient, mass conserving, optimally accurate, and able to handle complex geometries; therefore, this method has great potential to be useful for realistic problems involving coupled free flow and porous media flow. To offset the lack of the inf-sup condition of the P1–P0 and P1–P1 elements for the Stokes equation, a parameter free stabilization term is added to the discrete formulation. Stability and optimal error estimates are proved based on a bridge built up between the finite volume element method and the finite element method. An element level implementation of the stabilization term is discussed so that an existing code package can be conveniently modified to handle the stabilization procedures. A series of numerical experiments are provided to illustrate the above features of the proposed method, the theoretical results, and the realistic applications.

Analysis of Subgrid Stabilization Method for Stokes-Darcy Problems

A number of techniques, used as remedy to the instability of the Galerkin finite element formulation for Stokes like problems, are found in the literature. In this work we consider a coupled Stokes-Darcy problem, where in one part of the domain the fluid motion is described by Stokes equations and for the other part the fluid is in a porous medium and described by Darcy law and the conservation of mass. Such systems can be discretized by heterogeneous mixed finite elements in the two parts. A better method, from a computational point of view, consists in using a unified approach on both subdomains. Here, the coupled Stokes-Darcy problem is analyzed using equal-order velocity and pressure approximation combined with subgrid stabilization. We prove that the obtained finite element solution is stable and converges to the classical solution with optimal rates for both velocity and pressure.

A modified local and parallel finite element method for the mixed Stokes–Darcy model

Based on two-grid discretizations, a modified local and parallel finite element algorithm for the mixed Stokes–Darcy problem is proposed and analyzed in this paper. The idea is that, on the coarse grid, we solve a coupled Stokes–Darcy problem; on the fine grid, we first consider the discrete Darcy problem with a coarse grid approximation uH to the interface coupling conditions, then solve a series of independent local Stokes subproblems with the numerical solution of Darcy problem on fine grid to the interface coupling conditions in parallel. Optimal error estimates are obtained, and numerical experiments demonstrate the accuracy and efficiency of the modified local and parallel algorithm.