A Finite Element Variational Multiscale Method Based on Two Local Gauss Integrations for Stationary Conduction-Convection Problems
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.
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 , where the incompressible fluid is the Boussinesq approximation to the nonstationary Navier-Stokes equations. Christon et al.  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.  combined proper orthogonal decomposition (POD) with the Petrov-Galerkin least squares mixed finite element (PLSMFE) method for the problems. In , a Newton iterative mixed finite element method for the stationary conduction-convection problems was shown by Si et al. In , Si and He gave a defect-correction mixed finite element method for the stationary conduction-convection problems. In , an analysis of conduction natural convection conjugate heat transfer in the gap between concentric cylinders under solar irradiation was carried out. In , 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 , Hughes et al. [18–20] and Layton , 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 :
Lemma 2.1 (see ). 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 , 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
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.
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.
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.
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).
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.
The project is supported by NSF of China (10971164) and the Research Foundation of Xianyang Normal University (06xsyk265).
K. Chida, “Surface temperature of a flat plate of finite thickness under conjugate laminar forced convection heat transfer condition,” International Journal of Heat and Mass Transfer, vol. 43, no. 4, pp. 639–642, 1999.View at: Google Scholar
Z. D. Luo, The Bases and Applications of Mixed Finite Element Methods, Science Press, Beijing, China, 2006.
Z. D. Luo and X. M. Lu, “A least-squares Galerkin/Petrov mixed finite element method for stationary conduction-convection problems,” Mathematica Numerica Sinica, vol. 25, no. 2, pp. 231–244, 2003.View at: Google Scholar
M. A. Christon, P. M. Gresho, and S. B. Sutton, “Computational predictability of time-dependent natural convection flows in enclosures (including a benchmark solution),” International Journal for Numerical Methods in Fluids, vol. 40, no. 8, pp. 953–980, 2002.View at: Publisher Site | Google Scholar | Zentralblatt MATH
T. J. R. Hughes, “Multiscale phenomena: green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods,” Computer Methods in Applied Mechanics and Engineering, vol. 127, no. 1–4, pp. 387–401, 1995.View at: Google Scholar | Zentralblatt MATH