Decoupling the Stationary Navier-Stokes-Darcy System with the Beavers-Joseph-Saffman Interface Condition

and Applied Analysis 3 Q ch ⊂ Q c . We assume that X ch and Q ch satisfy the inf-sup condition [56, 57]

Recently the more physically valid Navier-Stokes-Darcy model has attracted scientists' attention, and several coupled finite element methods have been studied for it [46][47][48][49][50][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 Stoke-Darcy model [6, 10-18, 21, 22].In this paper, we will develop a domain decomposition method for the Navier-Stokes-Darcy model based on Robin boundary conditions constructed from the interface conditions.This physics-based 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 Navier-Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition.In Section 3, we recall the coupled weak formulation and the corresponding coupled finite element method for the Navier-Stokes-Darcy model.In Section 4, a parallel domain decomposition method and its finite element discretization are proposed to decouple the Navier-Stokes-Darcy system by using the Robin-type 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.

Stationary Navier-Stokes-Darcy Model
In this section we introduce the following coupled Navier-Stokes-Darcy model on a bounded domain Ω = Ω  ∪ Ω  ⊂ R  , ( = 2, 3); see Figure 1.In the porous media region Ω  , the flow is governed by the Darcy system Here, ⃗   is the fluid discharge rate in the porous media, K 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 second-order formulation, which eliminates ⃗   in the Darcy system: In the fluid region Ω  , the fluid flow is assumed to be governed by the Navier-Stokes equations: where ⃗   is the fluid velocity,   is the kinematic pressure, ⃗   is the external body force, ] is the kinematic viscosity of the fluid, T( ⃗   ,   ) = 2]D( ⃗   ) −   I is the stress tensor, and is the deformation tensor.Let Γ = Ω  ∩ Ω  denote the interface between the fluid and porous media regions.On the interface Γ, 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 Γ, respectively,   ( = 1, . . .,  − 1) denote mutually orthogonal unit tangential vectors to the interface Γ, and ∏ = K]/.The third condition (7) is referred to as the Beavers-Joseph-Saffman (BJS) interface condition [52][53][54][55].
In this paper, for simplification, we assume that the hydraulic head   and the fluid velocity ⃗   satisfy the homogeneous Dirichlet boundary condition except on Γ, that is,   = 0 on the boundary Ω  /Γ and ⃗   = 0 on the boundary Ω  /Γ.

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 Navier-Stokes-Darcy model with Beavers-Joseph-Saffman condition.Let (⋅, ⋅)  denote the  2 inner product on the domain  ( = Ω  or Ω  ) and let ⟨⋅, ⋅⟩ denote the  2 inner product on the interface Γ or the duality pairing between ( 1/2 00 (Γ))  and  1/2 00 (Γ) [5].Define the spaces the bilinear forms and the projection onto the tangent space on Γ: With these notations, the weak formulation of the coupled Navier-Stokes-Darcy model with BJS interface condition is given as follows [46][47][48][49][50][51] Assume that we have in hand regular subdivisions of Ω  and Ω  into finite elements with mesh size ℎ.Then one can define finite element spaces  ℎ ⊂   ,  ℎ ⊂   and  ℎ ⊂   .We assume that  ℎ and  ℎ satisfy the inf-sup condition [56,57] inf where  > 0 is a constant independent of ℎ.This condition is needed in order to ensure that the spatial discretizations of the Navier-Stokes equations used here are stable.See [56,57] for more details of finite element spaces  ℎ ,  ℎ , and  ℎ that satisfy (12).One example is the Taylor-Hood element pair that we use in the numerical experiments; for that pair,  ℎ consists of continuous piecewise quadratic polynomials and  ℎ consists of continuous piecewise linear polynomials.Then a coupled finite element method with Newton iteration for the coupled Navier-Stokes-Darcy model is given as follows [46]: find ( ⃗  ,ℎ ,  ,ℎ ,  ,ℎ ) ∈  ℎ ×  ℎ ×  ℎ in the following procedure.

Physics-Based Domain Decomposition Method
The coupled finite element method may end up with a huge algebraic system, which combines all parts from the Navier-Stokes equations, Darcy equation, and interface conditions together into one sparse matrix.Hence it is often impractical to directly apply this method to large-scale 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 Navier-Stokes and Darcy equations.
Let us first consider the following Robin condition for the Darcy system: for a given constant   > 0 and a given function   defined on Γ, Then, the corresponding weak formulation for the Darcy part is given by the following: for   ∈  2 (Γ), find φ ∈   such that Second, we can propose the following two Robin-type conditions for the Navier-Stokes equations: for a given constant   > 0 and given functions   and ⃗   defined on Γ, Then, the corresponding weak formulation for the Navier-Stokes equation is given by the following: for   ∈  2 (Γ), find ( ̂⃗   , p ) ∈   ×   such that Our next step is to show that, for appropriate choices of   ,  p ,   , 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 Navier-Stokes-Darcy system (11) For the necessity of the lemma, we pick  = 0 and ⃗ V such that   ⃗ V = 0 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 ( φ , ̂⃗   , p ) solves the coupled Navier-Stokes-Darcy system (11), which completes the proof.Now we use the decoupled weak formulation constructed above to propose a physics-based parallel domain decomposition method with Newton iteration as follows.(1) Initial values  0  and  0  are guessed.They may be taken to be zero.
(3)  +1  and  +1  are updated in the following manner: Then the corresponding domain decomposition finite element method is proposed as follows.
(1) Initial values  0 ,ℎ and  0 ,ℎ are guessed.They may be taken to be zero.

Numerical Example
We divide Ω  and Ω  into rectangles of height ℎ = 1/ and width ℎ, where  is a positive integer, and then subdivide each rectangle into two triangles by drawing a diagonal.The Taylor-Hood element pair is used for the Navier-Stokes equations, and the quadratic finite element is used for the second-order formulation of the Darcy equation.
For the coupled finite element method of the steady Navier-Stokes-Darcy model with BJS interface condition, These rates of convergence are consistent with the approximation capability of the Taylor-Hood element and quadratic element, which is third order with respect to  2 norm of ⃗   and   , second order with respect to  1 norm of ⃗   and   , and second order with respect to  2 norms of   .In particular, the third-order convergence rate of   observed above, which is better than the approximation capability of the linear element, is mainly due to the special analytic solution  = 0.
For the parallel DDM with ] = 1,  = 1,   = 0.3, and ℎ = 1/32, Figures 2 and 3 show the  2 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.
Then Tables 2 and 3 list some  2 errors in velocity, hydraulic head, pressure, and   for the parallel domain decomposition method with   = 0.3 and   = 1.2.The data in these two tables indicate the geometric convergence rate √  /  since all the error ratios are less than (√  /  ) 4 = (√1/4) 4 = 0.0625.

Conclusions
In this paper, a parallel physics-based domain decomposition method is proposed for the stationary Navier-Stokes-Darcy 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.

Figure 1 :
Figure 1: A sketch of the porous median domain Ω  , fluid domain Ω  , and the interface Γ.

Figure 2 :
Figure 2: Convergence for the velocity of the free flow (a) and the hydraulic head of the porous medium flow (b) versus the iteration counter  for the parallel DDM with BJS interface condition.

Figure 3 :
Figure 3: Convergence for the pressure of the free flow (a) and   (b) versus the iteration counter  for the parallel DDM with BJS interface condition.

Figure 4 :
Figure 4: Geometric convergence rate of the velocity of the free flow (a) and the hydraulic head of the porous medium flow (b) for the parallel DDM with BJS interface condition.

Figure 5 :
Figure 5: Geometric convergence rate of the pressure of the free flow (a) and   (b) versus the iteration counter  for the parallel DDM with BJS interface condition.

Table 1 :
Errors of the finite element method for the steady Navier-Stokes-Darcy model with BJS interface condition.

Table 2 :
2 errors in velocity and hydraulic head for the parallel DDM with BJS interface condition.

Table 3 :
2 errors in pressure and   for the parallel DDM with BJS interface condition.

Table 4 :
The iteration counter  versus the grid size ℎ for both the parallel Robin-Robin domain decomposition method with BJS interface condition.