Journal of Applied Mathematics

Volume 2012, Article ID 825609, 21 pages

http://dx.doi.org/10.1155/2012/825609

## Low-Order Nonconforming Mixed Finite Element Methods for Stationary Incompressible Magnetohydrodynamics Equations

^{1}Department of Mathematics, Zhengzhou University, Zhengzhou 450052, China^{2}College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China

Received 14 February 2012; Accepted 2 April 2012

Academic Editor: Livija Cveticanin

Copyright © 2012 Dongyang Shi and Zhiyun Yu. 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

The nonconforming mixed finite element methods (NMFEMs) are introduced and analyzed for the numerical discretization of a nonlinear, fully coupled stationary incompressible magnetohydrodynamics (MHD) problem in 3D. A family of the low-order elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field, and the magnetic field. The existence and uniqueness of the approximate solutions are shown, and the optimal error estimates for the corresponding unknown variables in -norm are established, as well as those in a broken -norm for the velocity and the magnetic fields. Furthermore, a new approach is adopted to prove the discrete Poincaré-Friedrichs inequality, which is easier than that of the previous literature.

#### 1. Introduction

This work deals with the numerical discretization of a nonlinear, fully coupled stationary incompressible MHD problem by a family of the low-order NMFEMs. This requires discretizing a system of partial differential equations that couples the incompressible Navier-Stokes equations with Maxwell's equations.

The MHD problem has a number of applications such as liquid-metal cooling of nuclear reactors, electromagnetic casting of metals, MHD power generation, and MHD ion propulsion (cf. [1, 2]). Thus, many studies have already been devoted to the MHD problem. For theoretical results, let us just mention those by [3–5]. It is important to employ effective numerical methods to approximate the exact solutions of the MHD problem because the exact solutions can be obtained only for some special cases [2]. Compared with the finite difference methods [6–8], most studies are performed by the finite element methods (FEMs) [9–24].

Precisely speaking, the work started with [9], where inf-sup stable mixed elements were used to discretize the velocity field and the pressure, and -conforming elements for the magnetic field, and the existence and uniqueness of the discrete solutions with inhomogeneous boundary condition satisfying certain assumptions were proved and the convergence analysis was presented. In contrast to the results of [9], [10] derived the same results without any restrictions on the boundary data of the velocity field. Reference [11] examined the long-term dissipativity and unconditional nonlinear stability of time integration algorithms for an incompressible MHD problem. Reference [12] dealt with a decoupled linear MHD problem involving electrically conducting and insulating regions by Lagrange finite elements and gave error estimates for a fully discrete scheme. For convex polyhedral domains, or domains with a boundary , the convergence analysis of a stabilized FEM, the optimal control method, and two-level FEMs were investigated in [13–15], [16], and [17], respectively.

On the other hand, some different approaches to achieve convergence results in general Lipschitz polyhedral domains were realized. For example, a mixed discrete formulation about the problem based on (curl)-conforming (edge) elements to approximate the magnetic field was proposed in [18, 19]. This observation motivated the works such as the least-squares mixed FEM used in [20], the mixed discontinuous Galerkin method employed in [21, 22], and the splitting method presented in [23, 24]. However, all the analyses in [9–24] are about the conforming FEMs except [22].

As we know, nonconforming FEMs have certain advantages over conforming FEMs in some aspects. Firstly, the nonconforming elements are much easier to be constructed to satisfy the discrete inf-sup condition. Secondly, nonconforming elements have been used effectively especially in fluid and solid mechanics due to their stability. We refer to [25–34] for more details on the properties of nonconforming elements applied to incompressible flow problems.

For the Stokes equations, [25, 26] considered the approximations of nonconforming element and the rotated element and got first-order accuracy, respectively. Reference [27] modified the rotated / element used in [26] and derived the same convergence order as [26]. For the Navier-Stokes equations, [28–30] obtained maximum norm estimates of element and the optimal error estimates of element both in broken -norm for the velocity field and in -norm for the pressure with moving grids and anisotropic meshes. Furthermore, NMFEMs also have been applied to other problems such as the Darcy-Stokes equations [31], the conduction-convection problem [32, 33], and the diffusion-convection-reaction equation [34].

Especially, [22] firstly presented a NMFEM with exactly divergence-free velocities for a incompressible MHD problem where the velocity and the magnetic fields were approximated by divergence-conforming elements and curl-conforming Nédélec elements, respectively, and derived nearly optimal error estimates. Motivated by the ideas of [22, 32, 34–36], in this paper, we are interested in discretizations for the MHD problem that are based on NMFEMs; a family of the low order elements will be adopted as approximation spaces for the velocity field, the piecewise constant element for the pressure, and the lowest order -conforming element for the magnetic field on hexahedra or tetrahedra. The existence and uniqueness of the approximate solutions are shown, and the optimal error estimates for the corresponding unknown variables in -norm are established, as well as those in a broken -norm for the velocity, and the magnetic fields. Furthermore, a new approach is adopted to prove the discrete Poincaré-Friedrichs inequality, which is easier than that of the previous literature [37, 38].

The organization of this paper is as follows. In Section 2, we introduce the mixed variational formulation for the MHD problem. Section 3 will give the nonconforming mixed finite element schemes. In Section 4, we state some important lemmas and prove the existence and uniqueness of the approximate solutions. In Section 5, the optimal error estimates for the pressure, the velocity and the magnetic fields in -norm are established, as well as ones in broken -norm for the velocity and the magnetic fields.

Throughout the paper, indicates a positive constant, possibly differs at different occurrences, which is independent of the mesh parameter , but may depend on and other parameters that appeared in this paper. Notations that are not especially explained are used with their usual meanings.

#### 2. Equations and the Mixed Variational Formulation

In this section, we will consider a nonlinear, fully coupled stationary incompressible MHD problem in 3D as follows (see [9, 15]).

*Problem (). * Find the velocity field , the pressure , the magnetic field such that
where is a simply connected, bounded domain with unit outward normal on . , , and are the Hartman number, interaction parameter, and magnetic Reynolds number, respectively. The symbols , and denote the Laplace, gradient, and divergence operators, respectively. . is the body force.

Set
here and later, .

The mixed variational formulation for Problem (**I**) is written as follows.

*Problem (). * Find , such that
where

It has been shown in [9, 37, 38] that for , , there hold

Let . For and , we will equip with the norm
respectively, where is the -norm.

The following result can be found in [9].

Theorem 2.1. * If , then Problem has at least a solution, in addition, that is unique provided that
**
and satisfying the stability bound
**
where , , and are positive constants only depending on the domain .*

#### 3. Nonconforming Mixed Finite Element Schemes

Let be regular and quasi-uniform tetrahedra or hexahedra partition of with mesh size . We use the finite element spaces , and to approximate the unknown variables , , and . The following assumptions about the space pair are provided:(A) for all , ;(B) a constant, ;(C) is a norm ;(D)for all , ;(E)for all , , , , where stands for the jump of across the face if is an internal face, and it is equal to itself if , is the interpolation operator associated with satisfying for , and is the polynomial space of degree less than or equal to one on .

Introduce the finite element space The finite element space is defined by where is a space of polynomials whose degrees for , , are equal to one. So these are the nonconforming mixed finite element schemes.

*Remark 3.1. *It can be checked that the nonconforming finite elements studied in [25–33, 39–45] satisfy the above assumptions (A)–(E).

#### 4. The Existence and Uniqueness of the Approximate Solutions and Some Lemmas

In this section, we will prove some lemmas and the existence and uniqueness of the discrete solutions of nonconforming mixed finite element approximations for MHD equations.

Let and the trilinear forms and the bilinear forms and be defined as follows:

for and , respectively.

Then the approximate formulation of Problem reads as follows.

*Problem (). *Find , such that for all , ,

From the definition of (4.3), satisfies the following antisymmetric properties [9]:

Let . For all , we define
respectively. Then it is easy to see that and are the norms over and is the norm over .

Lemma 4.1. *The following discrete Poincaré -Friedrichs inequality holds:
*

*Proof. *We consider the following problem:
Then by [3], the solution of (4.12) satisfies
On the one hand, by Green's formula and Hölder's inequality, we deduce that
Using (4.13)-(4.14) and choosing , we may get the desired result.

*Remark 4.2. *The method used in this lemma is different from and easier than that of [37, 38].

Lemma 4.3. *For , , and , we have*(1)*,
*(2)*,*(3)*. *

*Proof. *The first result is wellknown [30, 37, 38]. To prove the second result, we need the imbedding properties and the discrete imbedding inequality showed in [30, 32]:
Thus,
the assertion for is proved. The proof for is analogous.

Lemma 4.4. *Let , , and ; then the following results hold: *(1)*,
*(2)*,
*(3)*where are positive constants, independent of .*

*Proof. *Firstly, using the triangle inequality and Lemma 4.3 yields
Applying and the following inequality [9, 37, 38]
leads to
With the help of Hölder's inequality, we find
The proof is completed.

Lemma 4.5. * The spaces and satisfy the discrete inf-sup condition [37, 38]; that is, there exists such that
*

*Proof. *On the one hand, by [37, 38], there exists a constant such that
Therefore, by the assumption (E) and (4.22), we obtain
where . The proof is completed.

From Lemmas 4.4-4.5, we have the following.

Theorem 4.6. *For , Problem has at least one solution satisfying the stability bound . Moreover, Problem has a unique solution provided that .*

#### 5. The Convergence Analysis

In this section, we will state the main results of this paper, that is, the error estimates for the velocity and the magnetic fields in -norm.

Theorem 5.1. *Assume that
**
Let and be the solutions of Problems and , respectively. Then there hold *(1)*(2)**
where *

*Proof. *We proceed in two steps.*Step 1. *For , by Green's formula, we have
Thus,
Here, we have used the following equality:
On the other hand, we have from (4.8)
Subtraction of (4.8) from (5.6) yields
Let be an arbitrary element of , that is:
Then,
For all , by virtue of and (5.9), we get
Notice that
Let , and by (5.12), we obtain
Using the continuity properties of and the stability bounds for and in Theorems 2.1 and 4.6, respectively, the right-hand side of (5.14) can be bounded by
Next, the coercivity property of the form , continuity of in Lemma 4.4, stability bound for in Theorem 2.1, and the assumption allow us to bound the left-hand side of (5.14) as
Combining these bounds, we have
Then, applying the triangle inequality, we get
Now, for , taking the infimum of (5.18) yields
With the argument as [37], we know that
Substituting (5.20) into (5.19) implies (5.2).*Step 2. *For , we have from (5.9) that
Using the continuity properties of and and the discrete inf-sup condition (4.21) of Lemma 4.5, it follows that
Then, with the help of the triangle inequality and (5.2), we complete the proof.

Theorem 5.2. *Let , , , and be the solutions of Problems and , respectively. Then there holds
*

*Proof. *On the one hand, the interpolation theory gives
Therefore, by (5.24), we obtain
At the same time, for , we define the interpolation on each element as
Then there holds
On the other hand, by the similar techniques to [25–27, 29, 30, 32], we have
Substituting (5.24)–(5.28) into (5.2) and (5.3) yields the desired result.

Next, we will establish the error estimates in -norm for the velocity and the magnetic fields by use of the duality argument introduced in [46].

We consider the following dual problem. Find and such that.

The variational formulation of (5.29) is written as follows.

*Problem (). *Find and such that for all
Under the same hypotheses as Theorem 2.1, we may easily know that Problem has a unique solution .

We require that (5.29) be -regular, that is:
Let satisfy

Theorem 5.3. *Under the hypothesis of Theorem 5.2, let be the solution of Problem , and assume that (5.31) holds. Then we have
*

*Proof. *By (5.31) and (5.32), we deduce that
Multiplying and both sides of the first and the second equation of (5.29), respectively, and integrating by parts on each element, we see that
where
Subtraction of (5.9) yields
Note that
Now, setting in Problem , we have
From (5.35)–(5.39), we get
where
By (2.8), Lemma 4.4, and Theorem 4.6, we find
From [45], we know
By virtue of , we obtain
Let be a constant such that
Since , and (5.45), we obtain
Thus, by (5.31) and the approximation theory, there hold
Combining these inequalities and using Lemma 4.4 and the results from (5.39) to (5.46) yields the desired result.

*Remark 5.4. * The results obtained in this paper are also valid to the MHD equations with the following boundary conditions on when .

#### Acknowledgments

The research are supported by the National Natural Science Foundation of China (no. 10671184; no. 10971203), the National Science Foundation for Young Scientists of China (no. 11101384) and the Foundation and Advanced Technology Research Program of Henan Province, China (no. 122300410208).

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