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 conductionconvection problems.


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 , , 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 Ω be a bounded domain in R d d 2 or 3 , with Lipschitz-continuous boundary ∂Ω.In this paper, we consider the stationary conduction-convection problem as follows: x T represents the velocity vector, p x the pressure, T x the temperature, λ > 0 the Grashoff number, j 0, 1 T the two-dimensional vector and ν > 0 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.

Finite Element VMS Method
Here, we introduce some notations For h > 0, finite-dimension subspace X h , M h , W h ⊂ X, M, W is introduced which is associated with Ω e , a triangulation of Ω 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 τ h be the regular triangulations or quadrilaterals of the domain Ω and define the mesh parameter h max Ω e ∈τ h {diam Ω e }, where 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 d d o h in this scheme acts only on the small scales, φ i is the basis function of the velocity on the domain Ω such that its value is one at node x i and zero at other nodes, and N is the dimension of M h .The symmetric and positive matrices M k ij , k ≥ 2 and M 1 ij are the stiffness matrices computed by using k-order and 1-order Gauss integrations at element level, respectively.p i and q i , i 1, 2, . . ., N are the values of p h and q h at the node x i .In detail, the stabilized term can be rewritten as 2.6 L 2 -projection operator h : L 2 Ω → R 0 with the following properties 25 :

2.7
Lemma 2.1 see 26 .Let X h , M h be defined as above, then there exists a positive constant β independent of h, such that

2.8
Using the above notations, the VMS variational formulation of problems 1.1 reads as follows. Find where a u, v ν ∇u, ∇v , a T, ψ ∇T, ∇ψ d q, v q, div v , and

2.11
B 1 There exists a constant C which only depends on Ω, such that where ε is an arbitrary positive constant.

Stability Analysis
Theorem 3.2.Suppose that B 1 -B3 are valid and ε is a positive constant number, such that Proof.We prove this theorem by the inductive method.For m 1, 3.3 holds obviously.Assuming that 3.3 holds for m n − 1, we want to prove that it holds for m n.We estimate Δu n h firstly.Letting v h u n h , q h 0 in the first equation of 2.10 and using 2.13 , we get T 0 and using 2.14 , we have Cλ ∇T 0 0 .

3.5
Letting T n h k n h T 0 , ψ k n h in the second equation of 2.10 , we can obtain Using 2.12 , 2.14 , and 3.2 , we get

Error Analysis
In this section, we establish the H 1 -bound of the error u n h − u, T n h − T and L 2 -bounds of the error p n h − p. Setting e n , μ n , η n u n h − u h , p n h − p h , T n h − T h .Firstly, we give some Lemmas.
Lemma 4.1.In [4], are the solution of problem A 1 and A 2 , respectively, then there holds that

Lemma 4.2. Under the assumptions of Theorem 3.2, A 2 has a unique solution u h , p h , T h
The detail proof we can see 4, 13, 14 .
Theorem 4.3.Under the assumption of Theorem 3.2, there holds

4.3
Proof.Subtracting 2.10 from 2.9 , we get the following error equations, namely e n , μ n , η n satisfies

4.11
From the inductive method, we know, for n 1, subtracting 2.10 from 2.9 , we can get

4.12
Letting v h e 1 , q h μ 1 in 4.12 and using 2.14 , we have By 4.7 , we have

4.15
Letting q h 0 in 4.12 , 2.14 , and 3.9 , using Lemma 2.1, we get Assuming that 4.3 is true for n k − 1, using 4.7 and 4.11 , we know that both of them are valid for n k.Using 4.7 holds for n k, we let q h 0 in 4.4 and using Lemma 2.1, 4.5 , and 3.3 , we have 4.17

4.18
where F and H are two positive constants.
Proof.By using B 1 and triangle inequality, we have Combining 3.3 and 4.7 , we obtain

4.24
Here, we complete the proof.

4.25
holds, where F 1 and H are the positive constants.
Proof.By Lemma 4.1, Theorem 4.4, and the triangle inequality, this theorem is obviously true.

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

Convergence Analysis
In our experiment, Ω 0, 1 × 0, 1 is the unit square in R 2 .Let T 0 0 on left and lower boundary of the cavity, ∂T/∂n 0 on upper boundary of the cavity, and T 0 4y 1 − y on right boundary of the cavity see Figure 1 .Physics model of the cavity flows: t 0, that is, n 0 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 P 2 -P 1 -P 2 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.

Driven Cavity
In this experiment, Ω 0, 1 × 0, 1 is the unit square in R 2 .Let T 0 0 on left and lower boundary of the cavity, ∂T/∂n 0 on upper boundary of the cavity, and T 0 4y 1 − y on right boundary of the cavity see Figure 1 .Physics model of the cavity flows: t 0, that is, n 0 initial values on boundary.Solving the stationary conduction-convection equations by using the P 2 -P 1 -P 2 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 P 1 -P 1 -P 1 element agree completely with the "exact solution."In Figure 3, we choose Re 2000, divide the cavity into M × N 100 × 100, from left to right shows the numerical streamline, the numerical isobar, and the numerical isotherms.In Figure 4, we choose Re 3000, divide the cavity into M × N 100 × 100, 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 ε d 0.1 h is suitable for the Sobolev space.Throughout the paper, our analysis and numerical tests are all carried out for the P 1 -P 1 -P 1 element see Tables 1 and 2 .

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 P 1 -P 1 -P 1 and ε d 0.1 h agrees completely with the "exact solution," which shows that our method is highly efficient for the stationary conductionconvection problems.Numerical tests tell us that VMS finite element method based on two local Gauss integrations is very effective.

Lemma 3 . 1 .
The trilinear form b satisfies the following estimate:

Theorem 4 . 4 .
Under the assumptions of Theorem 4.3, then there holds that

10 Mathematical Problems in Engineering Theorem 4 . 5 .
Under the assumptions of Theorem 4.3, the following inequality: