Abstract

We present a new stabilized finite element method for incompressible flows based on Brezzi-Pitkäranta stabilized method. The stability and error estimates of finite element solutions are derived for classical one-level method. Combining the techniques of two-level discretizations, we propose two-level Stokes/Oseen/Newton iteration methods corresponding to three different linearization methods and show the stability and error estimates of these three methods. We also propose a new Newton correction scheme based on the above two-level iteration methods. Finally, some numerical experiments are given to support the theoretical results and to check the efficiency of these two-level iteration methods.

1. Introduction

In this paper, we consider steady Navier-Stokes equations with homogeneous Dirichlet boundary conditions: where is a bounded convex domain with 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.

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. Roughly speaking, the selected finite element spaces are required to satisfy the inf-sup condition, such as the finite element space constructed by the pair. However, from the computational cost point of view, the pair is of practical importance in scientific computation with the lower computational cost. Therefore, much attention has been attracted by the pair for simulating the incompressible flow. But, in this case, the inf-sup condition is not satisfied. A usual technique is to introduce the stabilized term in the finite element variational equation such that the inf-sup condition is enforced. There exist many stabilized methods, such as Brezzi-Pitkäranta stabilized method [1], locally stabilized method [2, 3], pressure stabilized method [4], stream upwind Petrov-Galerkin method [5], Douglas-Wang absolutely stabilized method [6], and pressure projection stabilized method [7, 8] and the references cited therein. Most of these stabilized methods necessarily introduce the stabilized parameters. Moreover, some of these methods are conditionally stable; that is, the stabilized parameters must satisfy some stable condition. Therefore, the development of stabilized methods free from stabilized parameters has become increasingly important.

In this paper, we combine the Brezzi-Pitkäranta stabilized method, which is unconditionally stable [9], with techniques of two-level discretizations to solve the numerical solution of the problem (1) under the uniqueness condition. Two-level 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 in the works of Xu [10, 11]. There exists a large amount of references about two-level finite element method for Navier-Stokes equations. For details, please see the works of An and Qiu [12], Ervin et al. [13], Franca and Nesliturk [14], de Frutos et al. [15, 16], Girault and Lions [17], Goswami and Damázio [18], He [19], He and Li [20], He and Wang [21], He et al. [22], Huang et al. [23], Layton [24], Layton and Tobiska [25], Li [26], Li and An [27, 28], Liu and Hou [29], and Zhu and Chen [30] and the references cited therein.

Based on the Brezzi-Pitkäranta stabilized finite element method, in this paper, we solve the nonlinear Navier-Stokes equations on the coarse mesh with mesh size in Step and then solve a linear system according to Stokes/Oseen/Newton iterative method on the fine mesh with mesh size in Step II. Denote by the finite element approximation solution on the fine mesh. If we suppose , then the error estimate derived is where is independent of and and the norms and are defined in the next section. It is obvious that if we choose , then two-level method discussed in this paper provides the same convergence order as the classical one-level method. Finally, we propose a Newton correction scheme on the fine mesh. The numerical solution in Step is as the iterative initial value. Then the finite element approximation solution is solved in terms of Newton iterative scheme on the fine mesh in Step . The error estimate derived for this Newton correction scheme is Thus, if , then this new two-level method also is of the same convergence order as the classical one-level method.

This paper is organized as follows. In Section 2, we introduce some function spaces and some classical results about Navier-Stokes equations. In Section 3, the Brezzi-Pitkäranta stabilized finite element approximation will be applied and the error estimates about the velocity in -norm and -norm and the pressure in -norm are derived. In Section 4, the two-level discretization finite element methods are proposed and the error estimates (2) and (3) are shown. In the final section, the numerical experiments are displaced to support the theoretical results.

2. Navier-Stokes Equations

In what follows, we employ the standard notation (or ), , for the Sobolev spaces of all functions having square integrable derivatives up to order in . Denote the standard Sobolev norm by . If , we write (or ) and instead of (or ) and , respectively. The symbol always denotes some positive constant which is independent of the mesh parameters and and can be a different constant even in the same formulation.

Introduce the following spaces usually used in this paper: The space is equipped with the norm It is well known that is equivalent to due to Poincaré inequality. Introduce the following bilinear and trilinear forms: It is easy to check that this trilinear form satisfies the following important properties [20, 31]: for all and for all , , and , where depends only on .

Given , under the above notations, the variational formulation of the problem (1) reads as follows: find such that for all Define a generalized bilinear form on by then the problem (11) also takes the following form:

The following existence, uniqueness, and regularity results concerning the solution to the problem (13) are classical [32–34].

Theorem 1. Assuming that and satisfy the following uniqueness condition: then the problem (13) exists a unique solution satisfying Furthermore, if is of class , then the solution to the problem (13) satisfies the following regularity property:

3. Stabilized Finite Element Approximation

Let be a family of quasiuniform triangular partitions of into triangles. The corresponding ordered triangles are denoted by . Let , , and . For every , let denote the space of the polynomials on of degree at most . Consider the conforming finite element spaces and given by Then the Brezzi-Pitkäranta stabilized finite element approximation of (11) is as follows: find and such that for all where the stabilized term is defined by with some positive constant . Define a mesh-dependent norm on by Then, it holds that for all and which has been shown by Latché and Vola [35]. Moreover, also is defined for any couple of functions and satisfies

Introduce another generalized bilinear form on defined by Then the discrete problem (18) can be rewritten as follows:

Denote by and the standard interpolation operators satisfying Moreover, we suppose that the inverse inequalities hold:

First, we recall the following stable theorem [9].

Theorem 2. For any , there exist two positive constants and independent of such that on satisfies the following continuous property: and the weakly coercive property:

A direct result of Theorem 2 is that the problem (24) exists a unique solution. In order to derive the error estimate between and , we introduce the following Galerkin projection operator defined by for each and all . According to Theorem 2, it is easy to check that is well defined. Moreover, there holds

About the Galerkin projection operator , the following approximation property has been derived in [9].

Theorem 3. For any and , there holds

Next, we begin to show the error estimate for the one-level finite element approximation solution .

Theorem 4. Suppose that the uniqueness condition (14) holds. If and are the solutions of (13) and (24), respectively, then, for any , the following optimal error estimate holds:

Proof. First, we estimate . Setting and in (24), using (7) and Young inequality, we obtain Then under the uniqueness condition (14), satisfies
It follows from (30) that According to (7), (15), (34), and Young inequality, we get Thus, from (31) we obtain Next, we estimate . It follows from (15), (28), (34), and (37) that Thus, we obtain Moreover,

Next, we give the error estimate by Aubin-Nitsche technique. This error analysis is based on the regularity assumption that the following linearized problem (41) is regular. Given , find such that for all According to (7) and (15), it is easy to verify that the problem (41) exists a unique solution . The assumption that (41) is regular means that also belongs to and the following estimate holds: Under the above assumption, we prove the following theorem.

Theorem 5. Suppose that the uniqueness condition (14) holds. If and are the solutions of (13) and (24), respectively, then, for any , the following optimal error estimate holds:

Proof. Setting in the first equation of (41), it yields Subtracting (11) from (18) yields Taking and in (45) and combining them with (44), we obtain Using (31), (32), and (42), is estimated by Similarly, is estimated by About , we rewrite it as Then it follows from (8), (15), (31), (32), and (42) that Finally, using (22), (31), (32), and (42) we estimate by Combining these estimates for to with (46), we complete the proof of (43).