About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 125139, 17 pages
http://dx.doi.org/10.1155/2013/125139
Research Article

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

College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China

Received 2 April 2013; Revised 12 June 2013; Accepted 19 June 2013

Academic Editor: Stanislaw Migorski

Copyright © 2013 Yuan Li and Rong An. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [24], 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 [1017]. 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