Abstract
We present a new stabilized finite element method for Navier-Stokes equations with friction slip boundary conditions based on Brezzi-Pitkäranta stabilized method. The stability and error estimates of numerical solutions in some norms are derived for standard one-level method. Combining the techniques of two-level discretization method, we propose two-level Newton iteration method and show the stability and error estimate. Finally, the numerical experiments are given to support the theoretical results and to check the efficiency of this two-level iteration method.
1. Introduction
Consider the steady incompressible flows governed by the following steady incompressible Navier-Stokes equations Here ia a bounded domain and is assumed to have Lipschitz continuous boundary . denotes the velocity vector of the flows, denotes the pressure, and denote the body force vector. The constant is the viscous coefficient. The solenoidal condition indicates that the flows are incompressible.
In this paper, we consider the following friction slip boundary conditions: where . is a scalar function. and are the normal and tangential components of the velocity. , independent of , is the tangential components of the stress vector which is defined by with . The set defined in next section denotes a subdifferential set of the function at .
Navier-Stokes equations (1) with friction boundary conditions (2) are introduced by Fujita [1] to describe some problems in hydrodynamics. Subsequently, some well-posedness problems about the solutions to the steady and nonstationary problems are studied by many scholars, such as Fujita [2–4], Y. Li and K. Li [5, 6], Saito and Fujita [7], Saito [8], and references cited therein.
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 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 [14–16]. From the computational cost point of view, the 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 two-level Stokes/Oseen/Newton iteration methods [18, 19], from which we observe that if the coarse mesh and the fine mesh 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 ( pair). However, from the computational cost point of view, the pair is of practical importance in scientific computation with the lower computational cost than the pair. Therefore, much attention has been attracted by the pair for simulating the incompressible flow. But the discrete inf-sup condition does not hold for 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. 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 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 . Denote the finite element approximation solution on the fine mesh. If we suppose that the solution to the problem (1) and (2) belongs to , then the error estimate derived is where is independent of and and the norms and are defined in next section. Thus, we show that if , 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 -norm and the pressure in -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 two-level method based on Uzawa iteration. In final section, the numerical experiments are displayed to support the theoretical results.
2. Navier-Stokes Equations with Friction Boundary Conditions
In what follows, we employ the standard notation and , , for the Sobolev spaces of all functions having square integrable derivatives up to order in and the standard Sobolev norm. In particular for , we write and instead of and , respectively. We use the boldface Sobolev spaces and to denote the vector Sobolev spaces and , respectively. Throughout this paper, the symbol always denotes some positive constant which is independent of the mesh parameter 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 if and only if
For the mathematical setting, we introduce the following function spaces usually used in this paper: We define the norm in by Then is equivalent to the standard Sobolev norm due to Poincaré inequality.
We also introduce the following continuous bilinear forms and on and , respectively, by and a trilinear form on by It is well known that from Korn's inequality that where and both are some positive constants. Moreover, it is easy to check that this trilinear form satisfies the following important properties [31, 32]: for all and for all , where depends only on .
Given and with on , based on the above notations, the variational formulation of the problem (1) and (2) reads as follows: find such that for all with , 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 such that for all , 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 with where satisfies
3. Stabilized Finite Element Approximation
In this section, we assume that is a convex polygon. Let be a quasiuniform family of 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 . The finite element spaces and are constructed by Then the Brezzi-Pitkäranta stabilized finite element approximation formulation of the problem (13) reads as follows: find such that for all with the stable parameter , where the stabilized term on is defined by Define a mesh-dependent norm on by Then, it holds that for all and which has been shown by Latché and Vola [33]. Moreover, also is defined for any couple of functions and satisfies
Now, we introduce a generalized bilinear form on by 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 with
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 such that
Define the following Galerkin projection operators and defined by for each and all . It is obvious that Moreover, the following approximation properties about the Galerkin projection operators and have been derived in [28]: for any and . In terms of the trace inequality , there holds
Based on the above assumptions and notations, the and error estimates for the velocity and pressure in one-level finite element approximation (25) are derived.
Theorem 2. Under the uniqueness condition (15), suppose that and are the solutions to the problems (13) and (25), respectively; then, one has the following error estimate:
Proof. It follows from (24) that
Taking and in the first inequality of (13) and adding them yielded
Substituting the above inequality into (33), we obtain
From Hölder inequality and Young inequality, can be estimated by
Similarly, and satisfy
where we use the inequality (22). We rewrite as
Then from (11), (16), and (26), it is estimated by
It follows from triangular inequality that satisfies
Finally, we estimate 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 and , taking and 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), (43), and inverse inequality that
Thus, we get from (46) and triangular inequality that
4. 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 with are two real positive parameter. The coarse mesh triangulation is made as like in Section 3. And a fine mesh triangulation is generated by a mesh refinement process to . The finite element space pairs and corresponding to the triangulations and , respectively, are constructed as in Section 3. With the above notations, we propose the following two-level Newton iteration scheme.
Step 1. We solve the problem (25) on the coarse mesh; that is, find such that for all
Step 2. We solve a linearized Navier-Stokes type variational inequality problem according to Newton iteration on the fine mesh; that is, find such that for all
In this section, we will assume that the following stable condition holds:
In this case, the problem (49) exists a unique solution with
Taking and in (50), respectively, it yields
Then from (11) and (52), the left-hand side of (53) satisfies
which implies that the problem (50) admits a unique solution . Moreover, it is easy to check that satisfies
On the other hand, according to Theorem 2, the finite element approximation solution on the coarse mesh satisfies the following error estimate:
The error estimate for the two-level Newton iteration scheme is derived in the following theorem.
Theorem 3. Under the stable condition (51), suppose that and are the solutions to the problems (13) and (50), respectively; then, for sufficiently small , one has the following error estimate:
Proof. Proceeding as in the proof of Theorem 2, we have
Moreover, the terms can be estimated, respectively, by
Since can be rewritten as
then from (11), (16), (55), and (56), all terms in the right-hand side of (60) are estimated, respectively, by
where we use under the condition (51) and
Combining (59)–(62) with (58), for sufficiently small such that , we get
Thus, we obtain from triangular inequality and (30), (31), and (56) that
Next, we show the error estimate for the pressure. For all and , taking and in (44) and (50), respectively, and subtracting them yielded
Since there holds
then it is easy to show that
where we use (27) and (65). Therefore, the estimates (30), (56), and (64) imply that
5. 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).
Step 1. Denote . and then we solve and with on the coarse mesh by where The condition of iteration stop is .
Step 2. Denote . and we solve and with on the fine mesh by The condition of iteration stop is .
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 (see Figure 1). The body force is determined by the first equation in (1).
It is easy to verify that the exact solution satisfies on and on . The tangential vector on and had been and . Thus, we select On the other hand, from the friction slip boundary conditions (2), there holds and then the function can be chosen as on and .
In all numerical experiments, the viscous coefficient and the stable parameter are chosen as and . The parameter in Uzawa iteration scheme is chosen as . According to Theorem 3, we choose ; then, two-level finite element approximation solution is of the following error estimate: Here we select eight fine mesh values . 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, for the velocity in -norm and the pressure in -norm. We also observe that if , 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 , two-level Newton iteration scheme can reach the theoretical convergence orders of for both velocity and pressure, in -norm and -norm, respectively. Besides, we find that the current method also achieves the predicted convergence order of for velocity in the sense of -norm. From the view of computational cost, we can obviously observe by comparing Tables 1 and 2 that two-level 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.
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Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is based upon work funded by the National Natural Science Foundation of China under Grant Nos. 10901122 and 11001205 and by Zhejiang Provincial Natural Science Foundation of China under Grant No. LY12A01015.