Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models
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Decoupling the Stationary NavierStokesDarcy System with the BeaversJosephSaffman Interface Condition
Abstract
This paper proposes a domain decomposition method for the coupled stationary NavierStokes and Darcy equations with the BeaversJosephSaffman interface condition in order to improve the efficiency of the finite element method. The physical interface conditions are directly utilized to construct the boundary conditions on the interface and then decouple the NavierStokes and Darcy equations. Newton iteration will be used to deal with the nonlinear systems. Numerical results are presented to illustrate the features of the proposed method.
1. Introduction
The StokesDarcy model has been extensively studied in the recent years due to its wide range of applications in many natural world problems and industrial settings, such as the subsurface flow in karst aquifers, oil flow in vuggy porous media, industrial filtrations, and the interaction between surface and subsurface flows [1–8]. Since the problem domain naturally consists of two different physical subdomains, several different numerical methods have been developed to decouple the Stokes and Darcy equations [6, 9–26]. For other works on the numerical methods and analysis of the StokesDarcy model, we refer the readers to [27–45].
Recently the more physically valid NavierStokesDarcy model has attracted scientists’ attention, and several coupled finite element methods have been studied for it [46–51]. On the other hand, the advantages of the domain decomposition methods (DDMs) in parallel computation and natural preconditioning have motivated the development of different DDMs for solving the StokeDarcy model [6, 10–18, 21, 22]. In this paper, we will develop a domain decomposition method for the NavierStokesDarcy model based on Robin boundary conditions constructed from the interface conditions. This physicsbased DDM is different from the traditional ones in the sense that they focus on decomposing different physical domains by directly utilizing the given physical interface conditions.
The rest of paper is organized as follows. In Section 2, we introduce the NavierStokesDarcy model with the BeaversJosephSaffman interface condition. In Section 3, we recall the coupled weak formulation and the corresponding coupled finite element method for the NavierStokesDarcy model. In Section 4, a parallel domain decomposition method and its finite element discretization are proposed to decouple the NavierStokesDarcy system by using the Robintype boundary conditions constructed from the physical interface conditions. Finally, in Section 5, we present a numerical example to illustrate the features of the proposed method.
2. Stationary NavierStokesDarcy Model
In this section we introduce the following coupled NavierStokesDarcy model on a bounded domain , (); see Figure 1. In the porous media region , the flow is governed by the Darcy system Here, is the fluid discharge rate in the porous media, is the hydraulic conductivity tensor, is a sink/source term, and is the hydraulic head defined as , where denotes the dynamic pressure, the height, the density, and the gravitational acceleration. We will consider the following secondorder formulation, which eliminates in the Darcy system:
In the fluid region , the fluid flow is assumed to be governed by the NavierStokes equations: where is the fluid velocity, is the kinematic pressure, is the external body force, is the kinematic viscosity of the fluid, is the stress tensor, and is the deformation tensor.
Let denote the interface between the fluid and porous media regions. On the interface , we impose the following three interface conditions: where and denote the unit outer normal to the fluid and the porous media regions at the interface , respectively, denote mutually orthogonal unit tangential vectors to the interface , and . The third condition (7) is referred to as the BeaversJosephSaffman (BJS) interface condition [52–55].
In this paper, for simplification, we assume that the hydraulic head and the fluid velocity satisfy the homogeneous Dirichlet boundary condition except on , that is, on the boundary and on the boundary .
3. Coupled Weak Formulation and Finite Element Method
In this section we will recall the coupled weak formulation and the corresponding coupled finite element method for the NavierStokesDarcy model with BeaversJosephSaffman condition. Let denote the inner product on the domain ( or ) and let denote the inner product on the interface or the duality pairing between and [5]. Define the spaces the bilinear forms and the projection onto the tangent space on :
With these notations, the weak formulation of the coupled NavierStokesDarcy model with BJS interface condition is given as follows [46–51]: find such that
Assume that we have in hand regular subdivisions of and into finite elements with mesh size . Then one can define finite element spaces , and . We assume that and satisfy the infsup condition [56, 57] where is a constant independent of . This condition is needed in order to ensure that the spatial discretizations of the NavierStokes equations used here are stable. See [56, 57] for more details of finite element spaces , , and that satisfy (12). One example is the TaylorHood element pair that we use in the numerical experiments; for that pair, consists of continuous piecewise quadratic polynomials and consists of continuous piecewise linear polynomials.
Then a coupled finite element method with Newton iteration for the coupled NavierStokesDarcy model is given as follows [46]: find in the following procedure.(1)The initial value is chosen.(2)For , solve (3)Set , , and .
4. PhysicsBased Domain Decomposition Method
The coupled finite element method may end up with a huge algebraic system, which combines all parts from the NavierStokes equations, Darcy equation, and interface conditions together into one sparse matrix. Hence it is often impractical to directly apply this method to largescale real world applications. In order to develop a more efficient numerical method in this section, we will directly utilize the three physical interface conditions to construct a physicsbased parallel domain decomposition method to decouple the NavierStokes and Darcy equations.
Let us first consider the following Robin condition for the Darcy system: for a given constant and a given function defined on , Then, the corresponding weak formulation for the Darcy part is given by the following: for , find such that
Second, we can propose the following two Robintype conditions for the NavierStokes equations: for a given constant and given functions and defined on ,
Then, the corresponding weak formulation for the NavierStokes equation is given by the following: for , find such that
Our next step is to show that, for appropriate choices of , , , and , the solutions of the coupled system (11) are equivalent to the solutions of the decoupled equations (15) and (17), and hence we may solve the latter system instead of the former.
Lemma 1. Let be the solution of the coupled NavierStokesDarcy system (11) and let be the solution of the decoupled NavierStokes and Darcy equations (15) and (17) with Robin boundary conditions at the interface. Then, if and only if , , , , and satisfy the following compatibility conditions:
Proof. Adding (15) and (17) together, we obtain the following: given , find such that
For the necessity of the lemma, we pick and such that in (11) and (20); then by subtracting (20) from (11), we get
which implies (19). The necessity of (18) can be derived in a similar fashion.
As for the sufficiency of the lemma, by substituting the compatibility conditions (18)(19), we easily see that solves the coupled NavierStokesDarcy system (11), which completes the proof.
Now we use the decoupled weak formulation constructed above to propose a physicsbased parallel domain decomposition method with Newton iteration as follows. (1) Initial values and are guessed. They may be taken to be zero.(2) For , independently solve the Darcy and NavierStokes equations constructed above. More precisely, is computed from and and are computed from the following Newton iteration.(i) Initial value is chosen for the Newton iteration. For instance, it may be taken to be and for .(ii) For , solve (iii) Set and .(3) and are updated in the following manner:
Then the corresponding domain decomposition finite element method is proposed as follows.(1) Initial values and are guessed. They may be taken to be zero. (2) For , independently solve the Darcy and NavierStokes equations with the Robin boundary conditions on the interface, which are constructed previously. More precisely, is computed from and and are computed from the following Newton iteration.(i) Initial value is chosen for the Newton iteration. For instance, it may be taken to be and for .(ii) For , solve (iii) Set and .(3) and are updated in the following manner:
5. Numerical Example
Example 1. Consider the model problem (2)–(6) with the BJS interface condition (7) on with and . Choose , , , , and , where is the identity matrix and . The boundary condition data functions and the source terms are chosen such that the exact solution is given by We divide and into rectangles of height and width , where is a positive integer, and then subdivide each rectangle into two triangles by drawing a diagonal. The TaylorHood element pair is used for the NavierStokes equations, and the quadratic finite element is used for the secondorder formulation of the Darcy equation.
For the coupled finite element method of the steady NavierStokesDarcy model with BJS interface condition, Table 1 provides errors for different choices of . Using linear regression, the errors in Table 1 satisfy These rates of convergence are consistent with the approximation capability of the TaylorHood element and quadratic element, which is third order with respect to norm of and , second order with respect to norm of and , and second order with respect to norms of . In particular, the thirdorder convergence rate of observed above, which is better than the approximation capability of the linear element, is mainly due to the special analytic solution .
For the parallel DDM with , , , and , Figures 2 and 3 show the errors of hydraulic head, velocity, pressure, and . We can see that the parallel domain decomposition method is convergent for . Moreover, Figures 4 and 5 show that a smaller leads to faster convergence.
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
Then Tables 2 and 3 list some errors in velocity, hydraulic head, pressure, and for the parallel domain decomposition method with and . The data in these two tables indicate the geometric convergence rate since all the error ratios are less than .


Finally, for the preconditioning feature of the domain decomposition method, Table 4 shows the number of iterations is independent of the grid size . Here, we set , , , and . Let , , and denote the finite element solutions of , , and at the th step of the domain decomposition algorithm. The criterion used to stop the iteration, that is, to determine the value , is , where the tolerance .
6. Conclusions
In this paper, a parallel physicsbased domain decomposition method is proposed for the stationary NavierStokesDarcy model with the BJS interface condition. This method is based on the Robin boundary conditions constructed from the three physical interface conditions. Moreover, it is convergent with geometric convergence rates if the relaxation parameter is selected properly. The number of iteration steps is independent of the grid size due to the natural preconditioning advantage of the domain decomposition methods.
Acknowledgments
This work is partially supported by DOE Grant DEFE0009843, National Natural Science Foundation of China (11175052).
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