Mathematical Problems in Engineering

VolumeΒ 2011, Article IDΒ 370192, 28 pages

http://dx.doi.org/10.1155/2011/370192

## A Defect-Correction Mixed Finite Element Method for Stationary Conduction-Convection Problems

Faculty of Science, Xi'an Jiaotong University, Xi'an 710049, China

Received 29 July 2010; Revised 15 November 2010; Accepted 5 January 2011

Academic Editor: Katica R. (Stevanovic)Β Hedrih

Copyright Β© 2011 Zhiyong Si and Yinnian He. 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 defect-correction mixed finite element method (MFEM) for solving the stationary conduction-convection problems in two-dimension is given. In this method, we solve the nonlinear equations with an added artificial viscosity term on a grid and correct this solution on the same grid using a linearized defect-correction technique. The stability is given and the error analysis in and -norm of , and the -norm of are derived. The theory analysis shows that our method is stable and has a good precision. Some numerical results are also given, which show that the defect-correction MFEM is highly efficient for the stationary conduction-convection problems.

#### 1. Introduction

In this paper, we consider the stationary conduction-convection problems in two dimension whose coupled equations governing viscous incompressible flow and heat transfer for the incompressible fluid are Boussinesq approximations to the stationary Navier-Stokes equations.

Find such that where is a bounded domain in assumed to have a Lipschitz continuous boundary . represents the velocity vector, the pressure, the temperature, the Grashoff number, the two-dimensional vector, and the viscosity.

As we know the conduction-convection problem contains the velocity vector field, the pressure field and the temperature field, so finding the numerical solution of conduction-convection problems is very difficult. The conduction-convection problems is an important system of equations in atmospheric dynamics and dissipative nonlinear system of equations, so lots of works are devoted to this problem [1β6]. There are also some works devoted to the nonstationary conduction-convection problems [7β10]. In [8], Luo et al. gave an optimizing reduced PLSMFE for the nonstationary conduction-convection problems. They combined PLSMEF method with POD to deal with the problems. In [11], an analysis of conduction natural convection conjugate heat transfer in the gap between concentric cylinders under solar irradiation was studied. In [12], a Newton iterative mixed finite element method for the stationary conduction-convection problems was shown by Si et al. In [13], Si and He gave a coupled Newton iterative mixed finite element method for the stationary conduction-convection problems.

The defect-correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Due to its good efficiency, there are many works devoted to this method, for example, [14β28]. In [18], a method making it possible to apply the idea of iterated defect correction to finite element methods was given. A method for solving the time-dependent Navier-Stokes equations, aiming at higher Reynolds' number, was presented in [23]. In [27], an accurate approximations for self-adjoint elliptic eigenvalues was presented. In [28], Stetter exposed the common structural principle of all these techniques and exhibit the principal modes of its implementation in a discretization context.

In this paper we present a defect-correction MFEM for the stationary conduction convection problems. In this method, we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct this solution on the same grid using a linearized defect-correction technique. Actually, the defect-correction MFEM incorporates the artificial viscosity term as a stabilizing factor, making both the nonlinear system easier to resolve and the linearized system easier to precondition. The stability and error analysis of the coupled the defect-correction MFEM show that this method is stable and has a good precision. Some numerical experiments show that our analysis is proper and our method is effective. And it can be used for solving the convection-conduction problems with much small viscosity.

This paper is organized as follows. In Section 2, the functional settings and some assumptions are given. Section 3 is devoted to the defect-correction MFEM. Section 4 gives the stability analysis. Section 5 presents the error analysis. In Section 6, some numerical results and the numerical analysis to validate the effectiveness of the method are laid out.

#### 2. Functional Setting for the Conduction Convection Problems

In this section, we aim to describe some of the notations and results which will be frequently used in this paper. The Sobolev spaces used in this context are standard [29]. For the mathematical setting of the conduction-convection problems and MFEM of conduction-convection problems (1.1), we introduce the Hilbert spaces is the uniformly regular family of triangulation of , indexed by a parameter . We introduce the finite element subspace , , as follows where is the space of piecewise polynomials of degree on , and , , are three integers. , and satisfies the discrete LBB condition where .

With the above notations, the Galerkin mixed variation and the mixed FEM problem for the conduction-convection problems are defined, respectively, as follows.

Find such that

Find such that where , , , and

The following assumptions and results are recalled (see [7, 29β31]).

(A_{1})There exists a constant which only depends on , such that(i), , for all ,(ii)for all ,(iii)for all.(A

_{2})Assuming , then, for , there exists an extension in (denote also), such that where is an arbitrary positive constant.(A

_{3}) and have the following properties.(i)For all , , there holds that (ii)For all, , for all, there holds that where

We recall the following existence, uniqueness and regularity result of (see [7, Chapter 4]).

Theorem 2.1 (see [7]). *Under the assumption of (A _{1})~(A_{3}), letting , , there exist , such that
*

*Then, there exists a unique solution for , and*

Some estimates of the trilinear form are given in the following lemma and the proof can be found in [30, 32β34].

Lemma 2.2. *The trilinear form satisfies the following estimate:
**
for all , .*

Lemma 2.3. *Suppose that (A _{1})~(A_{3}) are valid and is a positive constant, such that
*

*then has a unique solution , such that and*

*Proof. * The proof of the existence and the uniqueness of the solution has been given by Luo [7]. Let , in (2.5), we can get
Using (2.11) and (2.16), we deduce
Letting , in the first equation of (2.5), we get
By (2.16), we can obtian
Using (2.16) again, we get
By (2.19), we deduce

We introduce the Laplace operator

Lemma 2.4 (see [35, 36]). *For all , there holds that
*

#### 3. The Defect-Correction Method

The aim of this section is to give a method for solving the nonlinear system (2.5) on a coarser mesh than one uses when employing the standard FEM; the coarse-mesh solution is corrected using the same grid in our method. The defect-correction method in which we consider incorporates an artificial viscosity parameter as a stabilizing factor in the solution algorithm. For a fixed grid parameter the method requires the solution of one nonlinear system and a few linear correction steps. It is described in the following paragraphs. We consider the following problems which is identical to (2.5) except for an artificial viscosity term.

Find such that We define the residual or named defect , for the momentum systems as follows: Define the correction satisfying the following linear problem: Define , , , which are hoped to be better solutions of the problems. In order to obtain the equations for , we use the residual equation (3.2) to rewrite the linear problems (3.3); we obtain In general, this method can be described as follows.

*Step 1. *Solve the nonlinear systems (3.1) for .

*Step 2. *For , solve the linear equations
For each the residual is given by
The correction is given by

*Remark 3.1. *From the numerical experiments, we see that one or two correction steps is adequate. And this is as same as [24].

#### 4. Stability Analysis

In this section, we give the stability analysis. It is given by the following theorems.

Theorem 4.1. *Under the assumptions of Lemma 2.3, then defined by satisfies
**
Moreover, if
** admits a unique solution.*

*Proof. * We define the set

Let be in . Then
has a unique solution such that . For a given , we consider the following problem:
By the theory of the Navier-Stokes equations, we get (4.5) has a unique solution (see [31]). It means that (4.4) and (4.5) give a unique for a given , we denote

Setting , in (4.4) and using (2.9), we can obtain
Using (2.7), (2.11), and (2.16), we can get
Using the triangle inequality, we have

Letting , in (4.5) and using (2.8), we get
Letting and using (2.9), we have
Namely,
Hence, we proved that maps to . It follows from Brouwer's fixed-point theorem that there exits a solution to system .

To prove the uniqueness, assume that , and are two solutions of . Then, we obtain that
Let , in the first equation of (4.13), we can get
Setting in the second equation of (4.13), we obtain
By (4.14) and (4.15), we deduce
Using (4.2), we obtain
Namely,
By (4.15), we see that . Therefore, it follows that has a unique solution.

Then, we give the prove of (4.1) without using (4.2). Letting , in the first equation of (3.1) and using (2.8), we get
Letting , we have
Letting , in the second equation of (3.1) and using (2.9), we can obtain
Using (2.7), (2.11), and (2.16), we can get
By (4.20) and (4.22), we can deduce
Using (2.16), we get
Using (2.7), (2.11), (2.16), and (4.20), we can get
Therefore, we finish the proof.

Theorem 4.2. *Under the assumptions of Lemma 2.3, and
** defined by (3.4) satisfies
**
where . *

*Proof. * Letting , in the first equation of (3.4) and using (2.8), we get
Letting and using (2.10), we have
Let , in the second equation of (3.4), we can obtain
Using (2.11) and (2.16), we can get
Using (4.29), we get
Using (2.16), (4.26), and Theorem 4.2, we can obtain
Namely,

Using (2.16), (4.31), and (4.35), we can get
Using the triangle inequality, we can get
Therefore, we finish the proof.

#### 5. Error Analysis

In this section, we establish the and -bounds of the error , , and -bound of the error , . In order to obtain the error estimates, we define the Galerkin projection , such that

Lemma 5.1 (see [37, 38]). *The Galerkin projection satisfies
*

Lemma 5.2 (see [7]). *There exits for all holds that
**
When , there holds
**
There exits for all holds that
**
When , there holds
*

Lemma 5.3 (see [7]). *If (A _{1})~(A_{3}) hold and and are the solution of problem and , respectively, then there holds that
*

Lemma 5.4. *Under the assumptions of Lemma 2.3, is the solution of (3.1), defined by (3.4), then there hold
*

*Proof. * Subtracting (3.1) from (2.5) we get the error equations, namely satisfy
Letting in the first equation of (5.10) and using (2.11), (2.8), and (A_{1}), we can get
Hence, we deduce
Letting in the second equation of (5.10) and using (2.9), we obtain
Using (2.11), we can get
By (5.12), we deduce
Using (4.1), we can obtain
By using (2.16) and (2.17), there holds
Therefore, we can deduce
By (5.14) and (5.18), we can have
Letting , in the first equation of (5.10) and using (2.3), we have
Hence, we finish the proof.

Theorem 5.5. *Under the assumptions of Lemmas 2.3 and 5.3, the following inequality
**
holds, where is a positive constant numbers.*

*Proof. * By Lemmas 5.3, 5.4, and the triangle inequality this theorem is obviously true.

Lemma 5.6. *For all , , , there hold that
*

*Proof. * Letting , we have
Using (2.25), we can deduce (5.22). Because , (5.23) holds.

Theorem 5.7. *Under the assumptions of Lemmas 2.3 and 5.3, the following inequality:
**
holds, where is a positive constant.*

*Proof. * Subtracting (3.1) from (2.4) we get the error equations, namely,
Letting , , and using (5.1) and (5.3), we can get
Taking , in the first equation of (5.27), we obtain
Using (2.10) and (A_{1}), we deduce
Using Theorem 2.1, (2.16), (4.1), and (5.2), we can obtain
Taking in the second equation of (5.27) and using (2.9) we have
By (2.9), we have
Letting , and using Lemma 5.6, we can get
By assumption (A_{2}), letting and using Lemma 5.1 and (2.16), (4.26), and (5.33), we can deduce
Hence, we have

By (2.10) and (2.25), we can deduce
Using (5.29), we can obtain