Two-Level Iteration Penalty Methods for the Navier-Stokes Equations with Friction Boundary Conditions

and Applied Analysis 3 for all u, v,w ∈ V, and |b (u, v,w)| + |b (v, u,w)| + |b (w, u, v)| ≤ N‖u‖V‖v‖2 ‖w‖ , (15) for all u ∈ V, v ∈ H(Ω)2, and w ∈ L(Ω)2, where N > 0 depends only onΩ. Given f ∈ L(Ω)2 and g ∈ L(S) with g > 0 on S, under the above notation, the variational formulation of the problem (1)-(2) reads as follows: find (u, p) ∈ (V,M) such that for all (v, q) ∈ (V,M) a (u, v − u) + b (u, u, v − u) + j (v τ ) − j (u τ ) − d (v − u, p) ≥ (f , v − u) , d (u, q) = 0, (16) where j(η) = ∫ S g|η|ds for all η ∈ L(S)2. Saito in [8] showed that there exists some positive β > 0 such that β 󵄩󵄩󵄩󵄩q 󵄩󵄩󵄩󵄩 ≤ sup V∈V d (v, q) ‖v‖V ; (17) then the variational inequality (16) is equivalent to the following: find u ∈ V σ such that for all v ∈ V σ a (u, v − u) + b (u, u, v − u) + j (v τ ) − j (u τ ) ≥ (f , v − u) . (18) The existence and uniqueness theoremof the solutionu to the problem (18) has been shown in [19]. Here, we only recall it. Theorem 1. If the following uniqueness condition holds 4κ 1 N(‖f‖ + 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩L2(S) ) μ < 1, (19) then there exists a unique solution u ∈ V σ to the variational inequality problem (18) such that


Introduction
In this paper, we consider a two-level iteration penalty method for the incompressible flows which are governed by the incompressible Navier-Stokes equations: where Ω is a bounded domain in R 2 assumed to have a Lipschitz continuous boundary Ω, > 0 represents the viscous coefficient, u = ( 1 ( ), 2 ( )) denotes the velocity vector, = ( ) the pressure and f = ( 1 ( ), 2 ( )) the prescribed body force vector. The solenoidal condition div u = 0 means that the flows are incompressible.
Instead of the classical whole homogeneous boundary conditions, here we consider the following slip boundary conditions with friction type: u = 0, on Γ, where Γ ∩ = 0, Γ ∪ = Ω, and is a scalar function; u = u ⋅ n and u = u − u n are the normal and tangential components of the velocity, where n stands for the unit vector of the external normal to ; (u) = − n, independent of , is the tangential components of the stress vector which is defined by = (u, ) = ( (u) − ) with (u) = ( / ) + ( / ), , = 1, 2. The set (a) denotes a subdifferential of the function at a ∈ 2 ( ) 2 , whose definition will be given in the next section.
This type of boundary condition is firstly introduced by Fujita [1] where some problems in hydrodynamics are studied. Some theoretical problems are also studied by many scholars, such as Fujita in [2][3][4], Y. Li and K. Li [5,6], and Saito and Fujita [7,8] and references cited in their work.
The development of appropriate mixed finite element approximations is a key component in the search for efficient techniques for solving the problem (1) quickly and efficiently. Roughly speaking, there exist two main difficulties. One is the nonlinear term (u ⋅ ∇)u, which can be processed by the linearization method such as the Newton iteration method, Stokes iteration method, Oseen iteration method [9], or the two-level methods [10][11][12][13][14][15][16][17]. The other is that the velocity and the pressure are coupled by the solenoidal condition. The popular technique to overcome the second difficulty is to relax the solenoidal condition in an appropriate method and to result in a pseudocompressible system, such as the penalty method and the artificial compressible method [18]. Recently, using the Taylor-Hood element, the authors [19] study the penalty finite element method for the problem (1)- (2). Denote (u ℎ , ℎ ) as the penalty finite element approximation solution to (u, ) ∈ ( 3 (Ω) 2 , 2 (Ω)). The error estimate derived in [19] is where > 0 is the penalty parameter. However, the condition number of the numerical discretization for the penalty methods is ( −1 ℎ −2 ), which will result in an illconditioned problem when mesh size ℎ → 0. In order to avoid the choice of the small parameter , Dai et al. [20] have studied the iteration penalty finite element method and derive where ∈ N + is the iteration step number.
Throughout this paper, we will use to denote a positive constant whose value may change from place to place but that remains independent of ℎ, , and and that may depend on , Ω and the norms of u, , f, and .

Preliminary
First, we give the definition of the subdifferential property. Let be a given function possessing the properties of convexity and weak semicontinuity from below. We say that the set (a) is a subdifferential of the function at a ∈ 2 ( ) 2 if and only if In what follows, we employ the standard notation (Ω) (or (Ω) 2 ) and || ⋅ || , ≥ 0, for the Sobolev spaces of all functions having square integrable derivatives up to order in Ω and the standard Sobolev norm. When = 0, we write 2 (Ω) (or 2 (Ω) 2 ) and || ⋅ || instead of 0 (Ω) (or 0 (Ω) 2 ) and || ⋅ || 0 , respectively.
Given f ∈ 2 (Ω) 2 and ∈ 2 ( ) with > 0 on , under the above notation, the variational formulation of the problem (1)-(2) reads as follows: find (u, ) ∈ ( , ) such that for all (v, ) ∈ ( , ) where ( ) = ∫ | | for all ∈ 2 ( ) 2 . Saito in [8] showed that there exists some positive > 0 such that then the variational inequality (16) is equivalent to the following: find u ∈ such that for all v ∈ The existence and uniqueness theorem of the solution u to the problem (18) has been shown in [19]. Here, we only recall it.

Theorem 1. If the following uniqueness condition holds
then there exists a unique solution u ∈ to the variational inequality problem (18) such that

Iteration Penalty Finite Element Approximation
Suppose that Ω is a convex and polygon domain. Let T ℎ be a family of quasi-uniform 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 . For simplicity, we consider the conforming finite element spaces ℎ and ℎ defined by Denote 0ℎ = 0 ∩ ℎ . It is well known that 0ℎ and ℎ satisfy the Babuška-Brezzi condition [24,25]: where > 0 is a constant independent of ℎ. Denote ℎ and ℎ as the 2 orthogonal projections onto ℎ and ℎ , respectively, which satisfy It follows from the trace inequality ||v|| 2 ( ) ≤ ||v|| Let > 0 be some small parameter. The one-level iteration penalty finite element method for the problem (16) has been studied in [20], which can be described as follows.

Two-Level Iteration Penalty Methods
In this section, based on the iteration penalty method described in the previous section, the two-level iteration penalty finite element methods for (16) are proposed in terms of the Stokes iteration, Oseen iteration, or Newtonian iteration. From now on, and ℎ with ℎ < are two real positive parameters. The coarse mesh triangulation T is made as in Section 3. And a fine mesh triangulation T ℎ is generated by a mesh refinement process to T . The conforming finite element space pairs ( ℎ , ℎ ) and ( , ) ⊂ ( ℎ , ℎ ) corresponding to the triangulations T ℎ and T , respectively, are constructed as in Section 3. With the preavious notations, we propose the following two-level iteration finite element methods.

Two-Level Stokes Iteration Penalty Method.
In Steps 1 and 2, we solve (27) and (28) on the coarse mesh, as in the follwing. Step Step 2. In Step 3, we solve a Stokes-type variational inequality problem on the fine mesh in terms of the Stokes iteration, as in the following. 6 Abstract and Applied Analysis Step As a direct consequence of Theorem 2, the solution (u , ) to the problem (49) satisfies Next, we estimate u ℎ . Taking v ℎ = 0, ℎ = ℎ in (50), it yields That is, Suppose that the initial data satisfies then using (51)-(52), we can estimate ℎ by By the classical existence theorem for the variational inequality problem of the second kind in the finite dimension [27], we have the following. (55), there exists a unique solution (u ℎ , ℎ ) to the problem (50). Moreover, u ℎ satisfy (56).

Theorem 5. Under the uniqueness condition
It follows from Theorems 3 and 4 that (u , ) is of the following error estimates: Next, we begin to prove the following error estimate for the solution (u ℎ , ℎ ) to the problem (50).
Thus, we complete the proof of (59).

Two-Level Oseen Iteration Penalty Method.
In Steps 1 and 2, we solve (48) and (49) on the coarse mesh, as in the following.
In Step 3, we solve an Oseen type variational inequality problem on the fine mesh in terms of the Oseen iteration, as in the following.
Thus, we complete the proof of (80).

Two-Level Newton Iteration Penalty Method.
In Steps 1 and 2, we solve (48) and (49) on the coarse mesh, as in the following.
Step 2. For = 1, 2, . . ., find (u , ) ∈ ( , ) by (49). In Step 3, we solve a linearized Navier-Stokes type variational inequality problem on the fine mesh in terms of the Newton iteration, as in the following.
Thus, we complete the proof of (93).

An Improved Scheme.
In this section, we will propose a scheme to improve the error estimates derived in Theorems 6-8, which is described as follows.
In Steps 1 and 2, we solve (48) and (49) on the coarse mesh, as in the following.
At Step 4, we solve a Newton correction of (u ℎ , ℎ ) on the fine mesh in terms of Newton iteration, as in the following.
First, we show the following theorem.
Thus, set = 0.001 and 1/ℎ ≈ (1/ ) 9/5 . Tables 4 and 5 display the relative 1 errors of the velocity and the relative 2 errors of the pressure and their average convergence orders and CPU time when we use the twolevel Stokes iteration penalty method and two-level Oseen iteration penalty method, respectively. Based on Tables 4 and 5, the two-level Stokes/Oseen iteration penalty methods can reach the convergence orders of (ℎ 5/4 ) for both velocity and pressure, in 1 -and 2 -norms, respectively, as shown in (126).
(127)  Then we choose = 0.01 5/4 and 1/ℎ = (1/ ) 2 such that Because when = 1/16 and ℎ = 1/256, this method does not work and the computer displays "out of memory". Thus, in this experiment, we pick six coarse mesh size values; that is, = 1/4, 1/6, . . . , 1/14. Table 6 displays the relative 1 errors of the velocity and the relative 2 errors of the pressure and their average convergence orders and CPU time when we use the two-level Newton iteration penalty method. Based on Tables 4 and 5, we can see that the two-level Newton iteration penalty method also reaches the convergence orders of (ℎ 5/4 ) for both velocity and pressure, in 1 -and 2norms, respectively, as shown in (128). Figures 2, 3, 4, and 5 show the streamline of flow and the pressure contour of the numerical solution by the twolevel Stokes/Oseen/Newton iteration penalty methods and the exact solution, respectively.