Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 747391, 14 pages

doi:10.1155/2012/747391

Research Article

A Finite Element Variational Multiscale Method Based on Two Local Gauss Integrations for Stationary Conduction-Convection Problems

Yu Jiang,¹ Liquan Mei,1,2 Huiming Wei,³ Weijun Tian,1,4 and Jiatai Ge¹

¹School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

²Center for Computational Geosciences, Xi'an Jiaotong University, Xi'an 710049, China

³China Nuclear Power Simulation Technology Company Limited, Shenzhen 518115, China

⁴College of Mathematics and Information Science, Xianyang Normal University, Xianyang 712000, China

Received 6 July 2012; Revised 23 October 2012; Accepted 24 October 2012

Academic Editor: Hung Nguyen-Xuan

Copyright © 2012 Yu Jiang et al. 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.

A new finite element variational multiscale (VMS) method based on two local Gauss integrations is proposed and analyzed for the stationary conduction-convection problems. The valuable feature of our method is that the action of stabilization operators can be performed locally at the element level with minimal additional cost. The theory analysis shows that our method is stable and has a good precision. Finally, the numerical test agrees completely with the theoretical expectations and the “ exact solution,” which show that our method is highly efficient for the stationary conduction-convection problems.

1. Introduction

The conduction-convection problems constitute an important system of equations in atmospheric dynamics and dissipative nonlinear system of equations. Many authors have worked on these problems [1–8]. The governing equations couple viscous incompressible flow and heat transfer process [9], where the incompressible fluid is the Boussinesq approximation to the nonstationary Navier-Stokes equations. Christon et al. [10] summarized some relevant results for the fluid dynamics of thermally driven cavity. A multigrid (MG) technique was applied for the conduction-convection problems [11, 12]. Luo et al. [13] combined proper orthogonal decomposition (POD) with the Petrov-Galerkin least squares mixed finite element (PLSMFE) method for the problems. In [14], a Newton iterative mixed finite element method for the stationary conduction-convection problems was shown by Si et al. In [15], Si and He gave a defect-correction mixed finite element method for the stationary conduction-convection problems. In [3], an analysis of conduction natural convection conjugate heat transfer in the gap between concentric cylinders under solar irradiation was carried out. In [16], Boland and Layton gave an error analysis for finite element methods for steady natural convection problems. Variational multiscale (VMS) method which defines the large scales in a different way, namely, by a projection into appropriate subspaces, see Guermond [17], Hughes et al. [18–20] and Layton [21], and other literatures on VMS methods [22–24]. The new finite element VMS strategy requires edge-based data structure and a subdivision of grids into patches. It does not require a specification of mesh-dependent parameters and edge-based data structure, and it is completely local at the element level. Consequently, the new VMS method under consideration can be integrated in existing codes with very little additional coding effort.

For the conduction-convection problems, we establish such system that be a bounded domain in , with Lipschitz-continuous boundary . In this paper, we consider the stationary conduction-convection problem as follows:

where is the velocity deformation tensor, is a bounded convex domain. represents the velocity vector, the pressure, the temperature, the Grashoff number, the two-dimensional vector and the viscosity.

The study is organized as follows. In the next section, the finite element VMS method is given. In Section 3, we give the stability. The error analysis is given in Section 4. In Section 5, we show some numerical test. The last but not least is the conclusion given in Section 6.

2. Finite Element VMS Method

Here, we introduce some notations

For , finite-dimension subspace is introduced which is associated with , a triangulation of into triangles or quadrilaterals, assumed to be regular in the usual sense. In this study, the finite-element subspaces of personal preference are defined by setting the continuous piecewise (bi)linear velocity and pressure subspace, let be the regular triangulations or quadrilaterals of the domain and define the mesh parameter

where , is integers. if is triangular and if is quadrilateral. Here does not satisfy the discrete Ladyzhenskaya-Babuška-Brezzi (LBB) condition

Now, in order to stabilize the convective term appropriately for the higher Reynolds number and avoid the extra storage, we supply finite element VMS method that the local stabilization form of the difference between a consistent and an underintegrated mass matrices based on two local Gauss integrations at element level as the stabilize term

Here,

the stabilization parameter in this scheme acts only on the small scales, is the basis function of the velocity on the domain such that its value is one at node and zero at other nodes, and is the dimension of . The symmetric and positive matrices , and are the stiffness matrices computed by using -order and 1-order Gauss integrations at element level, respectively. and , are the values of and at the node . In detail, the stabilized term can be rewritten as

-projection operator with the following properties [25]:

Lemma 2.1 (see [26]). Let be defined as above, then there exists a positive constant independent of , such that

Using the above notations, the VMS variational formulation of problems (1.1) reads as follows.

Find such that

Given , find such that

where , and

There exists a constant which only depends on , such that

(i),(ii)(iii)

Assuming , then, for , there exists an extension in , such that

where is an arbitrary positive constant.

have the following properties.

(i) For all , there holds that

(ii) For all , there holds that

where  

3. Stability Analysis

Lemma 3.1. The trilinear form satisfies the following estimate:

Theorem 3.2. Suppose that are valid and is a positive constant number, such that

Then () defined by satisfies

Proof. We prove this theorem by the inductive method. For , (3.3) holds obviously. Assuming that (3.3) holds for , we want to prove that it holds for . We estimate firstly. Letting in the first equation of (2.10) and using (2.13), we get

Setting and using (2.14), we have

Letting in the second equation of (2.10), we can obtain

Using (2.12), (2.14), and (3.2), we get

Using (3.2), we have . Then,

Combining (2.12), (2.14), (3.2), and (3.6), we arrive at

Therefore, we finish the proof.

4. Error Analysis

In this section, we establish the - of the error and - of the error . Setting . Firstly, we give some Lemmas.

Lemma 4.1. In [4], If hold, ,  and are the solution of problem and , respectively, then there holds that

Lemma 4.2. Under the assumptions of Theorem 3.2, has a unique solution  , such that and

The detail proof we can see [4, 13, 14].

Theorem 4.3. Under the assumption of Theorem 3.2, there holds

Proof. Subtracting (2.10) from (2.9), we get the following error equations, namely satisfies

Here, let , in (4.5), then we have

By using (2.14), we get

In (4.4), we take , then

Using (2.13) and (2.14), we have

then, we obtain

By using . Equations (3.3) and (4.2), we get

From the inductive method, we know, for , subtracting (2.10) from (2.9), we can get

Letting in (4.12) and using (2.14), we have

then

By (4.7), we have

Letting in (4.12), (2.14), and (3.9), using Lemma 2.1, we get

Assuming that (4.3) is true for , using (4.7) and (4.11), we know that both of them are valid for . Using (4.7) holds for , we let in (4.4) and using Lemma 2.1, (4.5), and (3.3), we have

Theorem 4.4. Under the assumptions of Theorem 4.3, then there holds that

where and are two positive constants.

Proof. By using and triangle inequality, we have

Using Theorem 4.3, letting , we obtain (4.18). Taking in (4.4) and using (2.14), we get

By (2.14) and Lemma 3.1, we deduce

Combining (3.3) and (4.7), we obtain

Using (3.2), we get

Combining (3.2), (4.7), and (4.17), we get

Here, we complete the proof.

Theorem 4.5. Under the assumptions of Theorem 4.3, the following inequality:

holds, where and are the positive constants.

Proof. By Lemma 4.1, Theorem 4.4, and the triangle inequality, this theorem is obviously true.

5. Numerical Test

This section presents the numerical results that complement the theoretical analysis.

5.1. Convergence Analysis

In our experiment, is the unit square in . Let on left and lower boundary of the cavity, on upper boundary of the cavity, and on right boundary of the cavity (see Figure 1). Physics model of the cavity flows: , that is, initial values on boundary. In general, we cannot know the exact solution of the stationary conduction-convection equations. In order to get the exact solution, we design the procedure as follows. Firstly, solving the stationary conduction-convection equations by using the finite element pair, which holds stability, on the finer mesh, we take the solution as the exact solution. Secondly, the absolute error is obtained by comparing the exact solution and the finite element solutions with VMS methods. Finally, we can easily obtain errors and convergence rates.

(a)

(b)

(c)

Figure 1: From (a) to (c): physics model of the cavity flows, vertical midlines for = 2000, = 1/100, horizontal midlines for = 2000, = 1/100.

5.2. Driven Cavity

In this experiment, is the unit square in . Let on left and lower boundary of the cavity, on upper boundary of the cavity, and on right boundary of the cavity (see Figure 1). Physics model of the cavity flows: , that is, initial values on boundary. Solving the stationary conduction-convection equations by using the finite element pair, which holds stability results, on the finer mesh, we take the solution as the exact solution. From Figures 1 and 2, we know that the solution of finite element VMS using element agree completely with the “exact solution.” In Figure 3, we choose , divide the cavity into , from left to right shows the numerical streamline, the numerical isobar, and the numerical isotherms. In Figure 4, we choose , divide the cavity into , from left to right shows the numerical streamline, the numerical isobar, and the numerical isotherms.

(a)

(b)

Figure 2: From (a) to (b): vertical midlines for = 3000, = 1/100, horizontal midlines for = 3000, = 1/100.

(a)

(b)

(c)

Figure 3: For = 2000, = 1/100, from (a) to (c): velocity streamlines, the pressure level lines, numerical isotherms.

(a)

(b)

(c)

Figure 4: For = 3000, = 1/100, from (a) to (c): velocity streamlines, the pressure level lines, numerical isotherms.

Remark 5.1. Our VMS finite element method based on two local Gauss integrations and is suitable for the Sobolev space. Throughout the paper, our analysis and numerical tests are all carried out for the element (see Tables 1 and 2).

Table 1: VMS: element.

Table 2: VMS: element.

rate
rate
rate
rate
rate

10
/
/
/
/
/

20
1.9802
0.9666
1.8253
0.8523
0.7150

40
2.0202
0.9963
1.8617
0.8723
0.7721

60
2.0180
1.0028
1.8903
0.8934
0.7827

80
2.0300
1.0042
1.9033
0.8447
0.7806

6. Conclusion

In this paper, we studied a finite element VMS algorithm based on two local Gauss integrations to solve the stationary conduction-convection problem. From Figures 1 and 2, we see that the solution of VMS using and agrees completely with the “exact solution,” which shows that our method is highly efficient for the stationary conduction-convection problems. Numerical tests tell us that VMS finite element method based on two local Gauss integrations is very effective.

Acknowledgments

The project is supported by NSF of China (10971164) and the Research Foundation of Xianyang Normal University (06xsyk265).

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