Abstract
This paper presents two-level iteration penalty finite element methods to approximate the solution of the Navier-Stokes equations with friction boundary conditions. The basic idea is to solve the Navier-Stokes type variational inequality problem on a coarse mesh with mesh size in combining with solving a Stokes, Oseen, or linearized Navier-Stokes type variational inequality problem for Stokes, Oseen, or Newton iteration on a fine mesh with mesh size . The error estimate obtained in this paper shows that if , , and can be chosen appropriately, then these two-level iteration penalty methods are of the same convergence orders as the usual one-level iteration penalty method.
1. 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 is a bounded domain in assumed to have a Lipschitz continuous boundary , represents the viscous coefficient, denotes the velocity vector, the pressure and the prescribed body force vector. The solenoidal condition 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: where , and is a scalar function; and are the normal and tangential components of the velocity, where stands for the unit vector of the external normal to ; , independent of , is the tangential components of the stress vector which is defined by with . The set denotes a subdifferential of the function at , 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 [2–4], 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 , 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 [10–17]. 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 as the penalty finite element approximation solution to . The error estimate derived in [19] is where is the penalty parameter. However, the condition number of the numerical discretization for the penalty methods is , which will result in an ill-conditioned problem when mesh size . 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 is the iteration step number.
In this paper, we combine the iteration penalty method with the two-level method to approximate the solution of the problem (1)-(2). The iterative penalty method was first introduced by Cheng and Shaikh [21] for the Stokes equations and further used to solve the pure Neumann problem [22]. This iteration penalty method can be considered as the time discretization of the artificial compressible method [23]. The two-level iteration penalty methods studied in this paper can be described as follows. The first step and the second step are required to solve a small Navier-Stokes equations on the coarse mesh in terms of the iteration penalty method [20, 21]. The third step is required to solve a large linearization problem on the fine mesh in terms of the Stokes iteration, Oseen iteration, or Newtonian iteration, respectively. We prove that these two-level iteration penalty finite element solutions are of the following error estimate: Finally, we propose an improved correction iteration scheme for in terms of the Newton iteration method. We prove that the correction finite element solutions are of the following error estimates:
Throughout this paper, we will use to denote a positive constant whose value may change from place to place but that remains independent of , , and and that may depend on , and the norms of ,,, and .
2. 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 is a subdifferential of the function at if and only if
In what follows, we employ the standard notation (or ) and , , for the Sobolev spaces of all functions having square integrable derivatives up to order in and the standard Sobolev norm. When , we write (or ) and instead of (or ) and , respectively.
For the mathematical setting, we introduce the following spaces:
The space is equipped with the norm It is well known that is equivalent to due to Poincare's inequality. Introduce two bilinear forms and a trilinear form It is easy to verify that this trilinear form satisfies the following important properties [12, 23]: for all , and for all , and , where depends only on .
Given and with on , under the above notation, the variational formulation of the problem (1)-(2) reads as follows: find such that for all where for all . Saito in [8] showed that there exists some positive such that then the variational inequality (16) is equivalent to the following: find such that for all The existence and uniqueness theorem of the solution 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 to the variational inequality problem (18) such that where satisfies
3. Iteration Penalty Finite Element Approximation
Suppose that is a convex and polygon domain. Let be a family of quasi-uniform triangular partition of . The corresponding ordered triangles are denoted by . Let , and . For every , let denote the space of the polynomials on of degree at most . For simplicity, we consider the conforming finite element spaces and defined by Denote . It is well known that and satisfy the Babuška-Brezzi condition [24, 25]: where is a constant independent of . Denote and as the orthogonal projections onto and , respectively, which satisfy It follows from the trace inequality [26] that
Let 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.
Step 1. Find such that for all
Step 2. For , find such that for all
First, we give the a priori estimate of the solution to the problem (28).
Theorem 2. Suppose that is the solution to the problem (28); then it satisfies
Proof. Setting , in (27), using (12) and Young's inequality, it yields that Then we have For , setting , in (28), it yields that Thus, we obtain
The next theorem gives the error estimate between the solutions and to the problems (16) and (28), respectively. The proof can be found in [20].
Theorem 3. Let and be the solutions to the problems (16) and (28), respectively; then they satisfy
Next, we will show the error estimate for the penalty finite element approximation (28). This error analysis is based on the regularity assumption that the following linearized problem (35) is regular.
Given , find such that for all According to (12) and (20), it is easy to verify that there exists a unique solution to the problem (35). The assumption that (35) is regular means that also belongs to and the following inequality holds: Let be the orthogonal projections onto and satisfy
Theorem 4. Let and be the solutions to the problems (16) and (28), respectively; then they satisfy
Proof. Setting and in the first equation of (35), we get Taking in (16) and in (28), respectively, we obtain Subtracting them, we get Substituting the previous equation into (39), it yields that Using (34), (36), and (37), is estimated by Similarly, using (25), (34), (36), and (37), is estimated by We rewrite as Then, from (13), (20), (29), (34), (36), and (37), it holds that Finally, we estimate by Combining these estimates with (42), we conclude that (38) holds.
4. 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 with are two real positive parameters. The coarse mesh triangulation is made as in Section 3. And a fine mesh triangulation is generated by a mesh refinement process to . The conforming finite element space pairs and corresponding to the triangulations and , respectively, are constructed as in Section 3. With the preavious notations, we propose the following two-level iteration finite element methods.
4.1. 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 1. Find such that for all
Step 2. For , find such that for all
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.
Step 3. Find such that for all
As a direct consequence of Theorem 2, the solution to the problem (49) satisfies
Next, we estimate . Taking in (50), it yields
That is,
Suppose that the initial data satisfies
then using (51)-(52), we can estimate by
By the classical existence theorem for the variational inequality problem of the second kind in the finite dimension [27], we have the following.
Theorem 5. Under the uniqueness condition (55), there exists a unique solution to the problem (50). Moreover, satisfy (56).
It follows from Theorems 3 and 4 that is of the following error estimates: Next, we begin to prove the following error estimate for the solution to the problem (50).
Theorem 6. Suppose that the uniqueness condition (55) holds. Let and be the solutions to the problems (16) and (50), respectively; then they satisfy
Proof. Define a generalized bilinear form on by Taking in (50), we have Let and in the first inequality of (16); then Adding the above two inequalities gives Substituting the above inequality into (61), it yields that It follows from Hölder's inequality and Young's inequality that where is some small constant determined later. We rewrite as Then using (13), (15), and Young's inequality, we can estimate by It is easy to show that Finally, from triangle inequality, is estimated by Substituting (65)–(69) into (64), it yields that where we use (24)–(26) and (57)-(58). Next, we estimate . For all , let in (16) and in (50), respectively. Then we get Subtracting them and using (13), (15), we obtain Therefore, it follows from (23) that can be estimated by If we choose such that , then substituting (73) into (70), we show From (73), again, we obtain Thus, we complete the proof of (59).
4.2. Two-Level Oseen Iteration Penalty Method
In Steps 1 and 2, we solve (48) and (49) on the coarse mesh, as in the following.
Step 1. Find by (48).
Step 2. For , find by (49).
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.
Step 3. Find such that for all
From (12), it is easy to show that the solution to the problem (76) satisfies Suppose that the initial data satisfies then using (52), we can estimate by
For two-level Oseen iteration penalty method, the solution is of the following error estimate.
Theorem 7. Suppose that the uniqueness condition (78) holds. Let and be the solutions to the problems (16) and (76), respectively; then they satisfy
Proof. Proceeding as in the proof of (64), we can get In the above equation, , , , and have been estimated in the proof of Theorem 6. Here, we only estimate . Using (12), (13), (15), and Young's inequality, we have Then substituting (65), (68), (69), and (82) into (81), it yields that In (83), we use (24)–(26) and (57)-(58). Next, we estimate . For all , proceeding as in the proof of (72), from (51) and (78), we can show It follows from (23) and (84) that If we choose such that , then substituting (85) into (83), we show From (85), again, we obtain Thus, we complete the proof of (80).
4.3. 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 1. Find by (48).
Step 2. For , find 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.
Step 3. Find such that for all
In this section, we will suppose that the initial data satisfies Then from (51), satisfies . Let in (88). Using (12), we obtain Since then
Theorem 8. Suppose that the uniqueness condition (89) holds. Let and be the solutions to the problems (16) and (88), respectively; then they satisfy
Proof. Proceeding as the in proof of (64), we can get
In the above equation, , , , and have been estimated in the proof of Theorem 6. Here, we only estimate . We rewrite as
From (13), (20), and Young's inequality, is estimated by
Using (13) and (24), we estimate by
where is independent of , and . Similarly, it follows from (13), (24), and (57) that
where is independent of , and . Finally, we can estimate by
For sufficiently small , and such that , substituting (65), (68), (69), and (95)–(99) into (94), it yields that
For all , proceeding as in the proof of (72), we can show
Since
where is independent of , and , then from (23) we have
Thus, for sufficiently small , and such that
substituting (103) into (100), we obtain
From (103), again, we obtain
Thus, we complete the proof of (93).
Remark 9. In terms of Theorems 6, 7, and 8, if we choose for the two-level Stokes or Oseen iteration penalty methods and for the two-level Newton iteration penalty method, then
4.4. 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.
Step 1. Find by (48).
Step 2. For , find by (49).
At Step 3, we solve a linearized problem (50) or (76) or (88) on the fine mesh in terms of Stokes iteration or Oseen iteration or Newton iteration, as in the following.
Step 3. Find by (50) or (76) or (88).
At Step 4, we solve a Newton correction of on the fine mesh in terms of Newton iteration, as in the following.
Step 4. Find such that for all First, we show the following theorem.
Theorem 10. Let and be the solutions of (28) and (108), respectively. Then there holds that where is the solution to the problem (50) or (76) or (88).
Proof. Under the uniqueness condition (89), the solution satisfies . Taking in (28) and in (108) and adding them yield Using (13), Hölder's inequality, and Young's inequality, we obtain That is, For all , taking in the first inequality of (28) and in the first inequality of (108), it yields that Subtracting them and using (23), (112), we obtain
From (34) and Theorems 6–10, we get the following error estimates.
Theorem 11. Let and be the solutions to the problems (16) and (108), respectively. Then for the two-level Stokes or Oseen iteration penalty methods, they satisfy And for the two-level Newton iteration penalty method, they satisfy
Remark 12. If we choose , in (115) for two-level Stokes or Oseen iteration penalty methods and , in (116) for the two-level Newton iteration penalty method, then we obtain
5. Numerical Results
In this section, we will give numerical results to confirm the error analysis obtained in Section 4. Since these two-level Stokes/Oseen/Newton iteration penalty methods are given in the form of the variational inequality problems which are not directly solved, the appropriate iteration algorithm must be constructed. Here we use the Uzawa iteration algorithm introduced in [28].
For simplicity, we only give the Uzawa iteration method for solving the variational inequality problem (16). Similar schemes can be used to solve the two-level Stokes/Oseen/Newton iteration penalty schemes in Section 4. First, there exists a multiplier such that the variational inequality problem (16) is equivalent to the following variational identity problem: where . In this case, we can solve the problem (16) by the following Uzawa iteration scheme: then is known; we compute and by where
Consider the problems (1)-(2) in the fixed square domain (see Figure 1). Let . The external force is chosen such that the exact solution is
It is easy to verify that the exact solution satisfies on , on and on . Moreover, the tangential vector on and are and . Thus, we have On the other hand, from the nonlinear slip boundary conditions (2), there holds that then the function can be chosen as on and .
In all experiments, we choose , iteration initial value , and . In terms of Theorems 6 and 7, for the two-level Stokes/Oseen penalty iteration methods, there holds that Then we choose such that We pick eight coarse mesh size values; that is, . In Table 1, the scaling between and is given.
Setting , the comparison of relative error and for different is shown in Tables 2 and 3 when we use the two-level Stokes/Oseen penalty iteration methods with and . We can see that, for our present testing case, it suffices to set if it is hoped to be as large as possible.
Thus, set and . Tables 4 and 5 display the relative errors of the velocity and the relative errors of the pressure and their average convergence orders and CPU time when we use the two-level 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 for both velocity and pressure, in - and -norms, respectively, as shown in (126).
Next, we give the numerical results by using the two-level Newton iteration penalty method. In terms of Theorem 8, there holds that Then we choose and such that Because when and , 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, . Table 6 displays the relative errors of the velocity and the relative 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 for both velocity and pressure, in - and -norms, 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 two-level Stokes/Oseen/Newton iteration penalty methods and the exact solution, respectively.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant nos. 10901122, 11001205 and by Zhejiang Provincial Natural Science Foundation of China under Grant no. LY12A01015.