Two-Level Brezzi-Pitkäranta Discretization Method Based on Newton Iteration for Navier-Stokes Equations with Friction Boundary Conditions

and Applied Analysis 3 We define the norm in V by ‖k‖V = (∫ Ω |∇v|2dx) 1/2 , ∀v ∈ V. (6) Then ‖ ⋅ ‖ V is equivalent to the standard Sobolev norm ‖ ⋅ ‖ 1 due to Poincaré inequality. We also introduce the following continuous bilinear forms a(⋅, ⋅) and d(⋅, ⋅) on V × V and V ×M, respectively, by


Introduction
Consider the steady incompressible flows governed by the following steady incompressible Navier-Stokes equations Here Ω ⊂ R 2 ia a bounded domain and is assumed to have Lipschitz continuous boundary Ω.u = ( 1 ,  2 ) denotes the velocity vector of the flows,  denotes the pressure, and f = ( 1 ,  2 ) denote the body force vector.The constant  > 0 is the viscous coefficient.The solenoidal condition div u = 0 indicates that the flows are incompressible.
In this paper, we consider the following friction slip boundary conditions: u = 0, on Γ, u  = 0, −  (u) ∈      u      , on , where Γ ∩  = 0, Γ ∪  = Ω. is a scalar function.u  and u  are the normal and tangential components of the velocity.  (u) =  −   n, independent of , is the tangential components of the stress vector  which is defined by   =   (u, ) = (  (u) −   )  with   (u) =   /  +   /  , ,  = 1, 2. The set () defined in next section denotes a subdifferential set of the function  at  ∈  2 ().
Compared with Navier-Stokes equations with homogeneous Dirichlet boundary conditions, the variational formulation of the problem (1) and ( 2) is the variational inequality problem of the second kind.There exist some references about the finite element methods for solving the numerical solution of the problem (1) and (2).For example, using  1  −  1 element, Ayadi et al. studied the finite element approximation for Stokes problem and the error estimate derived is suboptimal [9].Kashiwabara obtained the optimal error estimate by defining the different numerical integration of the nondifferential term on the boundary  corresponding to the different finite element pairs [10,11].Djoko and Mbehou studied the direct finite element approximation for steady Stokes problem [12] and the fully discretization scheme for nonstationary Stokes problem [13].Li and An discussed the penalty and stabilized finite element approximation and corresponding two-level methods for the steady Navier-Stokes equations [14][15][16].From the computational cost point of view, the  1 −  1 element is of practical importance in scientific computation with the lower computational cost.Therefore, much attention has been attracted for simulating the incompressible flows.Y. Li and K. Li [17] applied the pressure projection stabilized method to solving the numerical solution of the problem (1) and (2).Subsequently, An and his collaborates studied the corresponding twolevel Stokes/Oseen/Newton iteration methods [18,19], from which we observe that if the coarse mesh  and the fine mesh ℎ are selected appropriately, then two-level iteration methods provide the same convergence rate as the standard one-level method.Moreover, CPU time can be largely saved.
On the other hand, in computational fluid dynamics, it is very important in searching the appropriate mixed finite element approximation to solve the numerical solutions of the problem (1) quickly and efficiently.Generally, the selected finite element spaces are required to satisfy the discrete inf-sup condition, such as the finite element space constructed by Taylor-Hood element ( 2 −  1 pair).However, from the computational cost point of view, the  1 −  1 pair is of practical importance in scientific computation with the lower computational cost than the  2 −  1 pair.Therefore, much attention has been attracted by the  1 −  1 pair for simulating the incompressible flow.But the discrete inf-sup condition does not hold for  1 −  1 pair.A usual technique is to introduce the stabilized term in the finite element variational equation.There exist many stabilized methods, such as Brezzi-Pitkäranta stabilized method [20], locally stabilized method [21,22], pressure stabilized method [23], stream upwind Petrov-Galerkin method [24], Douglas-Wang absolutely stabilized method [25], pressure projection stabilized method [26,27], and references cited therein.Most of these stabilized methods necessarily introduce the stabilized parameters and are conditionally stable.
In this paper, we combine the Brezzi-Pitkäranta stabilized method [20], which is unconditionally stable [28], with two-level discretization technique to approximate the problem (1) and (2) under a uniqueness condition.Twolevel discretization method has become a powerful tool in solving nonlinear partial differential equations.The basic idea is to capture "large eddies" by computing the initial approximation on the coarse mesh and then to obtain the fine approximation by solving a linearized problem corresponding to nonlinear partial differential equations on the fine mesh.More details can be referred to the works of Xu [29,30].Since the variational formulation of the problem (1) and ( 2) is of the form of variational inequality, in this paper, we solve nonlinear Navier-Stokes type variational inequality problem on the coarse mesh with mesh size  and solve a linearized Navier-Stokes type variational inequality problem corresponding to Newton iteration method on the fine mesh with mesh size ℎ.Denote (u ℎ ,  ℎ ) the finite element approximation solution on the fine mesh.If we suppose that the solution (u, ) to the problem (1) and (2) belongs to ( 2 (Ω) 2 ,  1 (Ω)), then the error estimate derived is where  > 0 is independent of ℎ and  and the norms ‖ ⋅ ‖  and ‖ ⋅ ‖ are defined in next section.Thus, we show that if  = (ℎ 1/2 ), then two-level method proposed in this paper provides the same convergence order as the usual one-level method.This paper is organized as follows.In Section 2, we introduce some function spaces and some theoretical results about the problem (1) and (2).In Section 3, the Brezzi-Pitkäranta stabilized finite element approximation will be applied and the error estimates about the velocity in  1norm and the pressure in  2 -norm are derived.In Section 4, the two-level Newton iteration method is proposed and the error estimate (3) is shown.In Section 5, we give the program implementation to solve the subproblems in twolevel method based on Uzawa iteration.In final section, the numerical experiments are displayed to support the theoretical results.

Navier-Stokes Equations with Friction Boundary Conditions
In what follows, we employ the standard notation   (Ω) and ‖ ⋅ ‖  ,  ≥ 0, for the Sobolev spaces of all functions having square integrable derivatives up to order  in Ω and the standard Sobolev norm.In particular for  = 0, we write  2 (Ω) and ‖ ⋅ ‖ instead of  0 (Ω) and ‖ ⋅ ‖ 0 , respectively.We use the boldface Sobolev spaces H  (Ω) and L 2 (Ω) to denote the vector Sobolev spaces   (Ω) 2 and  2 (Ω) 2 , respectively.Throughout this paper, the symbol  always denotes some positive constant which is independent of the mesh parameter ℎ,  and can be a different constant even in the same formulation.
First, we recall the definition of the subdifferential set.Let  be a given function which is of convexity and weak semicontinuity from below.The set () is a subdifferential of the function  at  ∈  2 () if and only if For the mathematical setting, we introduce the following function spaces usually used in this paper: Abstract and Applied Analysis 3 We define the norm in V by Then ‖ ⋅ ‖  is equivalent to the standard Sobolev norm ‖ ⋅ ‖ 1 due to Poincaré inequality.We also introduce the following continuous bilinear forms (⋅, ⋅) and (⋅, ⋅) on V × V and V × , respectively, by and a trilinear form on V × V × V by It is well known that from Korn's inequality that where  0 and  1 both are some positive constants.Moreover, it is easy to check that this trilinear form satisfies the following important properties [31,32]: for all u, v, w ∈ V and for all u ∈ V, v ∈ H 2 (Ω), w ∈ L 2 (Ω), where  > 0 depends only on Ω.
Given f ∈ L 2 (Ω) and  ∈  2 () with  > 0 on , based on the above notations, the variational formulation of the problem (1) and (2) reads as follows: find (u, ) ∈ V ×  such that for all (v, ) ∈ with (v  ) = ∫  |v  |, which is the variational inequality problem of the second kind with Navier-Stokes operator and is called Navier-Stokes type variational inequality problem.
Moreover, the variational inequality problem ( 13) is equivalent to the following: find u ∈ V  such that for all v ∈ V  , which is from the inf-sup condition derived by Saito [8].Now we recall the existence and uniqueness result about the solution to the problem ( 14) under the uniqueness condition (15), which has been shown by Y. Li and K. Li [17].
Theorem 1.If the following uniqueness condition holds: then the variational inequality problem (14) admits a unique solution u ∈ V  with where

Stabilized Finite Element Approximation
In this section, we assume that Ω is a convex polygon.Let T ℎ be a quasiuniform family of triangular partition of Ω.The corresponding ordered triangles are denoted by  1 ,  2 , . . .,   .Let ℎ  = diam(  ),  = 1, . . ., , and ℎ = max{ℎ 1 , ℎ 2 , . . ., ℎ  }.For every  ∈ T ℎ , let   () denote the space of the polynomials on  of degree at most .The finite element spaces V ℎ and  ℎ are constructed by Then the Brezzi-Pitkäranta stabilized finite element approximation formulation of the problem (13) reads as follows: find with the stable parameter  > 0, where the stabilized term  ℎ (⋅, ⋅) on  ℎ ×  ℎ is defined by Define a mesh-dependent norm [⋅] ℎ on  ℎ by Then, it holds that which has been shown by Latché and Vola [33].Moreover,  ℎ (, ) also is defined for any couple of functions ,  ∈  1 (Ω) and satisfies Now, we introduce a generalized bilinear form Then, in this case, the discrete problem ( 19) can be rewritten as follows: From the classical result for variational inequality problem of the second kind in finite dimension [34], it is easy to show, under the uniqueness condition (15), the problem ( 25) admits a unique solution To obtain the existence and uniqueness of the solution to the problem (25), we recall the stable result shown in [28]; that is, there exists some positive  > 0 such that Define the following Galerkin projection operators  ℎ : V → V ℎ and  ℎ :  →  ℎ defined by for each (w, ) ∈ V ×  and all (w ℎ ,  ℎ ) ∈ V ℎ ×  ℎ .It is obvious that Moreover, the following approximation properties about the Galerkin projection operators  ℎ and  ℎ have been derived in [28]: for any w ∈ H 2 (Ω) ∩ V and  ∈  1 (Ω) ∩ .In terms of the trace inequality ‖v‖  2 () ≤ ‖v‖ Based on the above assumptions and notations, the  1 and  2 error estimates for the velocity and pressure in onelevel finite element approximation (25) are derived.Theorem 2. Under the uniqueness condition (15), suppose that (u, ) ∈ H 2 (Ω) ∩ V ×  1 (Ω) ∩  and (u ℎ ,  ℎ ) ∈ V ℎ ×  ℎ are the solutions to the problems (13) and (25), respectively; then, one has the following error estimate: Proof.It follows from (24) that Taking v = u ℎ and v = 2u −  ℎ u in the first inequality of ( 13) and adding them yielded Substituting the above inequality into (33), we obtain From Hölder inequality and Young inequality,  1 can be estimated by Similarly,  3 and  5 satisfy where we use the inequality (22).We rewrite  2 as Then from (11), (16), and (26), it is estimated by It follows from triangular inequality that  4 satisfies Finally, we estimate  6 by Substituting ( 36)-( 41) into (35), we get which together with ( 23), ( 30), (31), and triangular inequality shows Next, we give the error estimate for the pressure.We rewrite (13) as For all w ℎ ∈ V 0ℎ and  ℎ ∈  ℎ , taking (v, ) = (u ± w ℎ ,  ±  ℎ ) and (v ℎ ,  ℎ ) = (u ℎ ± w ℎ ,  ℎ ±  ℎ ) in ( 44) and (19), respectively, and subtracting them yielded Then in terms of the stable result (27), there holds It follows from ( 11), ( 16), ( 23), ( 26), (30)

Two-Level Newton Iteration Method
In this section, based on Brezzi-Pitkäranta stabilized finite element approximation, the two-level Newton iteration methods for (13) are proposed.From now on,  and ℎ with ℎ <  < 1 are two real positive parameter.The coarse mesh triangulation T  is made as like in Section 3.And a fine mesh triangulation T ℎ is generated by a mesh refinement process to T  .The finite element space pairs (V ℎ ,  ℎ ) and (V  ,   ) ⊂ (V ℎ ,  ℎ ) corresponding to the triangulations T ℎ and T  , respectively, are constructed as in Section 3.
With the above notations, we propose the following two-level Newton iteration scheme.

Program Implementation
Since the subproblems (49) and ( 50) in two-level Newton iteration method both are nonlinear variational inequality problems, then the appropriate numerical iteration schemes are required.Here, we use Uzawa iteration method discussed by Y. Li and K. Li in [35], which is based on the following equivalence relationship.It is easy to show that Navier-Stokes type variational inequality problem ( 13) is equivalent to the following variational equation: where Then we use the following Uzawa iteration scheme to solve two-level Newton iteration scheme (49) and (50).
and then we solve (u   ,    ) and    with  ∈ N + on the coarse mesh by where The condition of iteration stop is ‖u   − u −1  ‖ < 10 −6 .
and we solve (u ℎ  ,  ℎ  ) and  ℎ  with  ∈ N + on the fine mesh by The condition of iteration stop is ‖u ℎ  − u ℎ −1 ‖ < 10 −6 .

Numerical Results
In this section, we give the numerical experiments to support the theoretical results derived in Sections 3 and 4. The testing example is quoted from [19]; namely, the exact solution is chosen as in the unit square Ω = (0, 1) × (0, 1) (see Figure 1).The body force f is determined by the first equation in (1).
It is easy to verify that the exact solution u satisfies u = 0 on Γ and u  = 0 on  =  1 ∪  2 .The tangential vector  on  1 and  2 had been (0, 1) and (−1, 0).Thus, we select On the other hand, from the friction slip boundary conditions (2), there holds and then the function  can be chosen as  = −  ≥ 0 on  1 and  2 .In all numerical experiments, the viscous coefficient and the stable parameter are chosen as  = 0.01 and  = 0.01.The parameter  in Uzawa iteration scheme is chosen as  = 0.5 .According to Theorem 3, we choose  = ℎ 1/2 ; then, two-level finite element approximation solution is of the following error estimate: Here we select eight fine mesh values ℎ = 1/4 2 , 1/6 2 , . . ., 1/18 2 .Then the corresponding coarse mesh values are obtained.These fine mesh values also are used in the numerical experiment for one-level finite element approximation.The numerical results are displayed in Tables 1 and 2, from which we observe the following conclusions.Based on Table 1, the numerical convergence orders reach the theoretical convergence orders derived in Theorem 2, namely, (ℎ 3/4 ) for the velocity in  1 -norm and the pressure in  2norm.We also observe that if ℎ = 1/18 2 , in this case, the standard one-level method cannot work and does not obtain the predicted numerical results.From Table 2, we can see that if  = ℎ 1/2 , two-level Newton iteration scheme can reach the theoretical convergence orders of (ℎ 3/4 ) for both velocity and pressure, in  1 -norm and  2 -norm, respectively.Besides, we find that the current method also achieves the predicted convergence order of (ℎ 7/4 ) for velocity in the sense of  2 -norm.From the view of computational cost, we can obviously observe by comparing Tables 1 and 2 that twolevel Newton iteration method significantly saves CPU time than one-level method and, meanwhile, obtains nearly the same approximation results.
Finally, we show the contour plots of the exact solution and the numerical solution to exhibit the approximation   profiles.Figures 2, 3, and 4 display the exact solution and the numerical solution by one-level method and two-level Newton method, respectively.From these three groups of contour plots, we can observe the good coincidence with each other to illustrate the stability of the present stabilized methods.

Figure 4 :
Figure 4: Contour plots of numerical solution by two-level Newton method.From (a) to (c): two components of velocity and pressure.

Table 1 :
Convergence of one-level method.

Table 2 :
Convergence of two-level Newton iteration scheme.