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.
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 [
Recently the more physically valid NavierStokesDarcy model has attracted scientists’ attention, and several coupled finite element methods have been studied for it [
The rest of paper is organized as follows. In Section
In this section we introduce the following coupled NavierStokesDarcy model on a bounded domain
A sketch of the porous median domain
In the fluid region
Let
In this paper, for simplification, we assume that the hydraulic head
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
With these notations, the weak formulation of the coupled NavierStokesDarcy model with BJS interface condition is given as follows [
Assume that we have in hand regular subdivisions of
Then a coupled finite element method with Newton iteration for the coupled NavierStokesDarcy model is given as follows [
The initial value
For
Set
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
Second, we can propose the following two Robintype conditions for the NavierStokes equations: for a given constant
Then, the corresponding weak formulation for the NavierStokes equation is given by the following: for
Our next step is to show that, for appropriate choices of
Let
Adding (
For the necessity of the lemma, we pick
As for the sufficiency of the lemma, by substituting the compatibility conditions (
Now we use the decoupled weak formulation constructed above to propose a physicsbased parallel domain decomposition method with Newton iteration as follows.
Initial values
For
and
Initial value
For
Set
Then the corresponding domain decomposition finite element method is proposed as follows.
Initial values
For
and
Initial value
For
Set
Consider the model problem (
For the coupled finite element method of the steady NavierStokesDarcy model with BJS interface condition, Table
Errors of the finite element method for the steady NavierStokesDarcy model with BJS interface condition.





































For the parallel DDM with
Convergence for the velocity of the free flow (a) and the hydraulic head of the porous medium flow (b) versus the iteration counter
Convergence for the pressure of the free flow (a) and
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.
Geometric convergence rate of the pressure of the free flow (a) and
Then Tables












0.0523 

0.0527 


0.0527 

0.0526 


0.0526 

0.0527 


0.0526 

0.0527 


0.0527 

0.0526 












0.0526 

0.0486 


0.0526 

0.0508 


0.0526 

0.0517 


0.0526 

0.0521 


0.0526 

0.0523 
Finally, for the preconditioning feature of the domain decomposition method, Table
The iteration counter










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.
This work is partially supported by DOE Grant DEFE0009843, National Natural Science Foundation of China (11175052).