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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 825609, 21 pages
http://dx.doi.org/10.1155/2012/825609
Research Article

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

1Department of Mathematics, Zhengzhou University, Zhengzhou 450052, China
2College 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 𝐿2-norm are established, as well as those in a broken 𝐻1-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 [35]. 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 [68], most studies are performed by the finite element methods (FEMs) [924].

Precisely speaking, the work started with [9], where inf-sup stable mixed elements were used to discretize the velocity field and the pressure, and 𝐻1-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 𝐶1,1, the convergence analysis of a stabilized FEM, the optimal control method, and two-level FEMs were investigated in [1315], [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 𝐻(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 [924] 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 [2534] 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 𝑃1/𝑃0 element and the rotated 𝑄1/𝑄0 element and got first-order accuracy, respectively. Reference [27] modified the rotated 𝑄1/𝑄0 element used in [26] and derived the same convergence order as [26]. For the Navier-Stokes equations, [2830] obtained maximum norm estimates of 𝑃1/𝑃0 element and the optimal error estimates of 𝐸𝑄1rot/𝑄0 element both in broken 𝐻1-norm for the velocity field and in 𝐿2-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, 3436], 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 𝐻1-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 𝐿2-norm are established, as well as those in a broken 𝐻1-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 𝐿2-norm are established, as well as ones in broken 𝐻1-norm for the velocity and the magnetic fields.

Throughout the paper, 𝐶 indicates a positive constant, possibly differs at different occurrences, which is independent of the mesh parameter , but may depend on Ω 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 (𝐈). Find the velocity field 𝑢=(𝑢1,𝑢2,𝑢3), the pressure 𝑝, the magnetic field 𝐵=(𝐵1,𝐵2,𝐵3) such that 𝑀2Δ𝑢+𝑁1𝑢𝑢+𝑝𝑅𝑚1𝑅(×𝐵)×𝐵=𝑓inΩ,𝑚1×(×𝐵)×(𝑢×𝐵)=0inΩ,𝑢=0inΩ,𝐵=0inΩ,𝑢=0on𝜕Ω,𝐵𝑛=0on𝜕Ω,(×𝐵)×𝑛=0on𝜕Ω,(2.1) where Ω is a simply connected, bounded domain with unit outward normal 𝑛=(𝑛1,𝑛2,𝑛3) on 𝜕Ω. 𝑀, 𝑁, and 𝑅𝑚 are the Hartman number, interaction parameter, and magnetic Reynolds number, respectively. The symbols Δ,, and denote the Laplace, gradient, and divergence operators, respectively. ×(×𝐵)=(𝐵)Δ𝐵. 𝑓𝐻1(Ω)3 is the body force.
Set 𝐻10(Ω)3=𝑣𝐻1(Ω)3;𝑣|𝜕Ω,𝐿=020(Ω)=𝑞𝐿2(Ω);Ω,𝐻𝑞𝑑𝐱=01𝑛(Ω)3=𝑣𝐻1(Ω)3;(𝑣𝑛)|𝜕Ω,=0(2.2) here and later, 𝐱=(𝑥,𝑦,𝑧).

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

Problem (𝐈1). Find (𝑢,𝐵)𝑊(Ω), 𝑝𝐿20(Ω) such that 𝑎((𝑢,𝐵),(𝑢,𝐵),(𝑣,Ψ))+𝑏((𝑣,Ψ),𝑝)=𝐹((𝑣,Ψ)),(𝑣,Ψ)𝑊(Ω),𝑏((𝑢,𝐵),𝜒)=0,𝜒𝐿20(Ω),(2.3) where 𝑊(Ω)=𝐻10(Ω)3×𝐻1𝑛(Ω)3,𝑎((𝑢,𝐵),(𝑣,Ψ),(𝑤,Φ))=𝑎0((𝑣,Ψ),(𝑤,Φ))+𝑎1𝑎((𝑢,𝐵),(𝑣,Ψ),(𝑤,Φ)),0((𝑣,Ψ),(𝑤,Φ))=𝑀2Ω𝑣𝑤𝑑𝐱+𝑅𝑚2Ω[]𝑎(×Ψ)(×Φ)+(Ψ)(Φ)𝑑𝐱,1((𝑢,𝐵),(𝑣,Ψ),(𝑤,Φ))=𝑐0(𝑢;𝑣,𝑤)𝑐1(𝐵;𝑤,Ψ)+𝑐2𝑐(𝐵;𝑣,Φ),0(𝑢;𝑣,𝑤)=Ω(2𝑁)1(𝑐𝑢𝑣𝑤𝑢𝑤𝑣)𝑑𝐱,1(𝐵;𝑤,Ψ)=Ω𝑅𝑚1𝑐(×Ψ)×𝐵𝑤𝑑𝐱,2(𝐵;𝑣,Φ)=Ω𝑅𝑚1(×Φ)×𝐵𝑣𝑑𝐱,𝑏((𝑣,Ψ),𝜒)=Ω𝜒𝑣𝑑𝐱,𝐹((𝑣,Ψ))=Ω𝑓𝑣𝑑𝐱.(2.4)
It has been shown in [9, 37, 38] that for 𝑢,𝑣,𝑤𝐻10(Ω)3, 𝐵,Ψ,Φ𝐻1𝑛(Ω)3, there hold 𝑐0(𝑢;𝑣,𝑤)=𝑐0(𝑢;𝑤,𝑣),𝑐0𝑎(𝑤;𝑣,𝑣)=0,1((𝑢,𝐵),(𝑣,Ψ),(𝑤,Φ))=𝑎1𝑎((𝑢,𝐵),(𝑤,Φ),(𝑣,Ψ)),1((𝑢,𝐵),(𝑣,Ψ),(𝑣,Ψ))=0.(2.5)
Let 𝑍(Ω)={𝑣𝐻10(Ω)3,𝑣=0}. For 𝑣𝐻10(Ω)3 and Ψ𝐻1𝑛(Ω)3, we will equip 𝑊(Ω) with the norm (𝑣,Ψ)𝑊=𝑣21+Ψ211/2,𝑓1=sup(0,0)(𝑣,Ψ)𝑊(Ω)𝑓((𝑣,Ψ))(𝑣,Ψ)𝑊,(2.6) respectively, where 1 is the 𝐻1-norm.

The following result can be found in [9].

Theorem 2.1. If 𝑓𝐻1(Ω)3, then Problem (𝐈1) has at least a solution, in addition, that is unique provided that 𝐶2𝛾3𝐶1𝛾12𝑓1<1(2.7) and satisfying the stability bound (𝑢,𝐵)𝑊𝐶1𝛾11𝑓1,(2.8) where 𝛾1=min{𝑀2,𝑅𝑚2}, 𝛾2=max{𝑀2,𝑅𝑚2}, 𝛾3=max{𝑁1,𝑅𝑚1} and 𝐶1,𝐶2 are positive constants only depending on the domain Ω.

3. Nonconforming Mixed Finite Element Schemes

Let Γ={𝐾} be regular and quasi-uniform tetrahedra or hexahedra partition of Ω with mesh size . We use the finite element spaces 𝑋1̸𝐻10(Ω)3, 𝑀𝐿20(Ω) and 𝑋2𝐻1𝑛(Ω)3 to approximate the unknown variables 𝑢, 𝑝, and 𝐵. The following assumptions about the space pair (𝑋1,𝑀) are provided:(A) for all 𝐾Γ, 𝑃1(𝐾)3𝑋1;(B)𝑀={𝜒𝐿20(Ω);𝜒|𝐾 a constant, 𝐾Γ};(C)1=(𝐾Γ||21,𝐾)1/2 is a norm 𝑋1;(D)for all 𝑣𝑋1,𝐹[𝑣]𝑑𝑠=0, 𝐹𝜕𝐾;(E)for all 𝑣𝐻10(Ω)3, 𝑞𝑀, 𝑏1(𝑣Π1𝑣,𝑞)=0, Π1𝑣1𝐶|𝑣|1, where [𝑣] stands for the jump of 𝑣 across the face 𝐹 if 𝐹 is an internal face, and it is equal to 𝑣 itself if 𝐹𝜕Ω, Π1 is the interpolation operator associated with 𝑋1 satisfying Π𝐾=Π1|𝐾 for 𝐾Γ, and 𝑃1(𝐾) is the polynomial space of degree less than or equal to one on 𝐾.

Introduce the finite element space 𝑅1𝑃(𝐾)=1𝑄(𝐾)if𝐾istetrahedra,1(𝐾)if𝐾ishexahedra.(3.1) The finite element space 𝑋2 is defined by 𝑋2=Ψ𝐻1𝑛(Ω)3;𝑞||𝐾𝑅1(𝐾)3,Ψ||𝑛𝜕Ω=0,𝐾Γ,(3.2) where 𝑄1(𝐾) is a space of polynomials whose degrees for 𝑥, 𝑦, 𝑧 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 [2533, 3945] 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 𝑊=𝑋1×𝑋2 and the trilinear forms 𝑎,𝑎1,𝑐𝑖(𝑖=0,1,2) and the bilinear forms 𝑎0 and 𝑏 be defined as follows:

for (𝑢,𝐵),(𝑣,Ψ),(𝑤,Φ)𝑊 and 𝜒𝑀,𝑎𝑢,𝐵,𝑣,Ψ,𝑤,Φ=𝑎0𝑣,Ψ,𝑤,Φ+𝑎1𝑢,𝐵,𝑣,Ψ,𝑤,Φ,(4.1)𝑎0𝑣,Ψ,𝑤,Φ=𝐾Γ𝑀2𝐾𝑣𝑤+𝑅𝑚2𝐾×Ψ×Φ+ΨΦ𝑑𝐱,(4.2)𝑎1𝑢,𝐵,𝑣,Ψ,𝑤,Φ=𝑐0𝑢;𝑣,𝑤𝑐1𝐵;𝑤,Ψ+𝑐2𝐵;𝑣,Φ,(4.3)𝑐0𝑢;𝑣,𝑤=𝐾Γ𝐾(2𝑁)1𝑢𝑣𝑤𝑢𝑤𝑣𝑑𝐱,(4.4)𝑐1𝐵;𝑤,Ψ=𝐾Γ𝐾𝑅𝑚1×Ψ×𝐵𝑤𝑑𝐱,(4.5)𝑐2𝐵;𝑣,Φ=𝐾Γ𝐾𝑅𝑚1×Φ×𝐵𝑣𝑑𝐱(4.6)𝑏𝑣,Ψ,𝜒=𝐾Γ𝐾𝜒𝑣𝑑𝐱,(4.7) respectively.

Then the approximate formulation of Problem (𝐈1) reads as follows.

Problem (𝐈2). Find (𝑢,𝐵)𝑊, 𝑝𝑀 such that for all (𝑣,Ψ)𝑊, 𝜒𝑀, 𝑎𝑢,𝐵,𝑢,𝐵,𝑣,Ψ+𝑏𝑣,Ψ,𝑝𝑣=𝐹,Ψ,𝑏𝑢,𝐵,𝜒=0.(4.8)
From the definition of (4.3), 𝑎1 satisfies the following antisymmetric properties [9]: 𝑎1𝑢,𝐵,𝑣,Ψ,𝑣,Ψ𝑎=0,1𝑢,𝐵,𝑣,Ψ,𝑤,Φ=𝑎1𝑢,𝐵,𝑤,Φ,𝑣,Ψ.(4.9)
Let 𝑍={𝑣𝑋1,𝑏((𝑣,Ψ),𝜒)=0}. For all 𝑣=(𝑣1,𝑣2,𝑣3)𝑋1,Ψ=(Ψ1,Ψ2,Ψ3)𝑋2, we define 𝑣0=𝐾Γ𝑣20,𝐾1/2,𝑣1=𝐾Γ||𝑣||21,𝐾1/2,(𝑣,Ψ)=𝐾Γ𝑣21+Ψ211/2,𝑓=sup(0,0)(𝑣,Ψ)𝑋1×𝑋2𝐹𝑣,Ψ(𝑣,Ψ),(4.10) respectively. Then it is easy to see that 0 and 1 are the norms over 𝑋1 and (,) is the norm over 𝑊.

Lemma 4.1. The following discrete Poincaré -Friedrichs inequality holds: Ψ0𝐶×Ψ0,Ψ𝑋2.(4.11)

Proof. We consider the following problem: 𝐵𝐵××=𝑓inΩ,𝐵=0inΩ,𝐵𝑛=0on𝜕Ω,××𝑛=0on𝜕Ω.(4.12) Then by [3], the solution 𝐵 of (4.12) satisfies 𝐵×0𝐶𝑓0.(4.13) On the one hand, by Green's formula and Hölder's inequality, we deduce that ||||Ω𝑓Ψ||||=|||||𝑑𝐱𝐾Γ𝐾𝐵××Ψ|||||𝐵𝑑𝐱×0×Ψ0.(4.14) Using (4.13)-(4.14) and choosing 𝑓=Ψ, 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 (𝑢,𝐵), (𝑣,Ψ), and (𝑤,Φ)𝑊, we have(1)|𝑐0(𝑢;𝑣,𝑤)|𝐶𝑢1𝑣1𝑤1, (2)|𝑐1(𝐵;𝑤,Ψ)|𝐶×Ψ0𝐵1𝑤1,(3)|𝑐2(𝐵;𝑣,Φ)|𝐶×Φ0𝐵1𝑣1.

Proof. The first result is wellknown [30, 37, 38]. To prove the second result, we need the imbedding properties 𝐻1𝑛(Ω)3𝐻1(Ω)3𝐿4(Ω)3 and the discrete imbedding inequality showed in [30, 32]: 𝑣0,2𝑘,Ω𝑣𝐶1,𝑣𝑋1,𝑘=1,2.(4.15) Thus, ||𝑐1𝐵;𝑤,Ψ||𝐾Γ𝐾1𝑅𝑚||×Ψ×𝐵𝑤||1𝑑𝐱𝑅𝑚(×Ψ)0𝐵0,4𝑤0,4𝐶(×Ψ)0𝐵1𝑤1,(4.16) the assertion for 𝑐1 is proved. The proof for 𝑐2 is analogous.

Lemma 4.4. Let (𝑢,𝐵), (𝑣,Ψ), and (𝑤,Φ)𝑊; then the following results hold: (1)|𝑎1((𝑢,𝐵),(𝑣,Ψ),(𝑤,Φ))|𝐶𝑐𝛾3(𝑢,𝐵)(𝑣,Ψ)(𝑤,Φ), (2)|𝑎0((𝑢,𝐵),(𝑢,𝐵))|𝐶𝑎𝛾1(𝑢,𝐵)2, (3)|𝑎0((𝑢,𝐵),(𝑣,Ψ))|𝐶𝛾2(𝑢,𝐵)(𝑣,Ψ),where 𝐶𝑐,𝐶𝑎 are positive constants, independent of .

Proof. Firstly, using the triangle inequality and Lemma 4.3 yields ||𝑎1𝑢,𝐵,𝑣,Ψ,𝑤,Φ||||𝑐0𝑢;𝑣,𝑤||+||𝑐1𝐵;𝑤,Ψ||+||𝑐2𝐵;𝑣,Φ||𝐶𝑐𝛾3(𝑢,𝐵)(𝑣,Ψ)(𝑤,Φ).(4.17) Applying 𝐻1𝑛(Ω)3𝐻1(Ω)3 and the following inequality [9, 37, 38] 𝑣0𝐶×𝑣0+𝑣0,𝑣𝐻1𝑛(Ω)3(4.18) leads to 𝑎0𝑢,𝐵,𝑢,𝐵=𝐾Γ𝑀2𝐾𝑢𝑢+𝑅𝑚2𝐾×𝐵×𝐵+𝐵𝐵𝑑𝐱=𝑀2𝑢20+𝑅𝑚2×𝐵20+𝐵20𝐶𝑎𝑀min2,𝑅𝑚2𝑢21+𝐵21=𝐶𝑎𝛾1(𝑢,𝐵)2.(4.19) With the help of Hölder's inequality, we find ||𝑎0𝑢,𝐵,𝑣,Ψ||𝐾Γ𝑀2𝐾||𝑢𝑣||+𝑅𝑚2𝐾||×𝐵×Ψ||+||𝐵Ψ||𝑀𝑑𝐱2𝑢0𝑣0+𝐶𝑅𝑚2×𝐵0×Ψ0+𝐵0Ψ0𝑀𝐶max2,𝑅𝑚2(𝑢,𝐵)(𝑣,Ψ)=𝐶𝛾2(𝑢,𝐵)(𝑣,Ψ).(4.20) The proof is completed.

Lemma 4.5. The spaces 𝑋1 and 𝑀 satisfy the discrete inf-sup condition [37, 38]; that is, there exists 𝛽>0 such that inf𝜒𝑀sup(𝑣,Ψ)𝑊𝑏𝑣,Ψ,𝜒(𝑣,Ψ)𝜒0𝛽.(4.21)

Proof. On the one hand, by [37, 38], there exists a constant 𝛽>0 such that inf𝜒𝐿20(Ω)sup(𝑣,Ψ)𝑊(Ω)𝑏((𝑣,Ψ),𝜒)(𝑣,Ψ)𝑊𝜒0𝛽.(4.22) Therefore, by the assumption (E) and (4.22), we obtain sup(𝑣,Ψ)𝑊𝑏𝑣,Ψ,𝜒(𝑣,Ψ)sup(𝑣,Ψ)𝐻10(Ω)3×𝑋2𝑏Π1𝑣,Ψ,𝜒(Π1𝑣,Ψ)=sup(𝑣,Ψ)𝐻10(Ω)3×𝑋2𝑏𝑣,Ψ,𝜒(Π1𝑣,Ψ)1𝐶sup(𝑣,Ψ)𝐻10(Ω)3×𝑋2𝑏𝑣,Ψ,𝜒(𝑣,Ψ)𝛽𝜒0,(4.23) where 𝛽=𝛽/𝐶>0. The proof is completed.

From Lemmas 4.4-4.5, we have the following.

Theorem 4.6. For 𝑓𝐻1(Ω)3, Problem (𝐈2) has at least one solution ((𝑢,𝐵),𝑝)𝑊×𝑀 satisfying the stability bound (𝑢,𝐵)(𝐶𝑎𝛾1)1f. Moreover, Problem (𝐈2) has a unique solution provided that 𝐶𝑐𝛾3(𝐶𝑎𝛾1)2𝑓<1.

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 𝐻1-norm.

Theorem 5.1. Assume that 𝐶𝑐𝛾3𝑓1𝐶𝑎𝐶1𝛾21<12.(5.1) Let ((𝑢,𝐵),𝑝)𝑊(Ω)×𝐿20(Ω) and ((𝑢,𝐵),𝑝)𝑊×𝑀 be the solutions of Problems (𝐈1) and (𝐈2), respectively. Then there hold (1)(𝑢,𝐵)(𝑢,𝐵)𝐶inf𝑣,Ψ𝑊(𝑣𝑢,𝐵),Ψ+inf𝑠𝑀𝑝𝑠0+sup𝑣,Ψ𝑍×𝑋2||𝐸𝑣,Ψ||𝑣,Ψ,(5.2)(2)𝑝𝑝0𝐶inf𝑣,Ψ𝑊𝑣(𝑢,𝐵),Ψ+inf𝑠𝑀𝑝𝑠0+sup𝑣,Ψ𝑊||𝐸𝑣,Ψ||𝑣,Ψ,(5.3) where 𝐸𝑣,Ψ=𝐾Γ𝜕𝐾𝑀2𝜕𝑢𝑣𝜕𝑛𝑝𝑣𝑛(2𝑁)1(𝑢𝑛)𝑢𝑣𝑑𝑠.(5.4)

Proof. We proceed in two steps.Step 1. For (𝑣,Ψ)𝑊, by Green's formula, we have 𝑎0𝑣(𝑢,𝐵),,Ψ+𝑎1𝑣(𝑢,𝐵),(𝑢,𝐵),,Ψ+𝑏𝑣,Ψ𝑣,𝑝𝐹,Ψ=𝐾Γ𝐾𝑀2𝑢𝑣𝑑𝐱+𝑅𝑚2𝐾(×𝐵)×Ψ𝑑𝐱+(2𝑁)1𝐾𝑢𝑢𝑣𝑢𝑣𝑢𝑑𝐱𝑅𝑚1𝐾(×𝐵)×𝐵𝑣×Ψ×𝐵𝑢𝑑𝐱𝐾𝑝𝑣𝑑𝐱𝐾𝑓𝑣=𝑑𝐱𝐾Γ𝐾𝑀2Δ𝑢𝑣𝑑𝐱+𝜕𝐾𝑀2𝜕𝑢𝑣𝜕𝑛+𝑑𝑠𝐾𝑅𝑚2×(×𝐵)Ψ𝑑𝐱+𝜕𝐾𝑅𝑚2(×𝐵×𝑛)Ψ+𝑑𝑠𝐾𝑁1𝑢𝑢𝑣𝑑𝐱𝜕𝐾(2𝑁)1(𝑢𝑛)𝑢𝑣𝑑𝑠𝑅𝑚1𝐾(×𝐵)×𝐵𝑣𝑑𝐱𝑅𝑚1𝐾(×𝑢×𝐵)Ψ+𝑑𝐱𝜕𝐾(𝑢×𝐵×𝑛)Ψ𝑑𝑠+𝐾𝑝𝑣𝑑𝐱𝜕𝐾𝑝𝑣𝑛𝑑𝑠𝐾𝑓𝑣=𝑑𝐱𝐾Γ𝐾𝑀2Δ𝑢+𝑁1𝑢𝑢+𝑝𝑅𝑚1(×𝐵)×𝐵𝑓𝑣+𝑅𝑚1𝐾𝑅𝑚1×(×𝐵)×(𝑢×𝐵)Ψ+𝑑𝐱𝜕𝐾𝑀2𝜕𝑢𝑣𝜕𝑛(2𝑁)1(𝑢𝑛)𝑢𝑣𝑝𝑣𝑣𝑛𝑑𝑠=𝐸,Ψ.(5.5) Thus, 𝑎0𝑣(𝑢,𝐵),,Ψ+𝑎1𝑣(𝑢,𝐵),(𝑢,𝐵),,Ψ+𝑏𝑣,Ψ𝑣,𝑝=𝐹,Ψ𝑣+𝐸,Ψ.(5.6) Here, we have used the following equality: Ω(×Φ)Ψ𝑑𝐱=𝜕Ω(Φ×𝑛)Ψ𝑑𝑠+ΩΦ(×Ψ)𝑑𝐱.(5.7) On the other hand, we have from (4.8) 𝑎0𝑢,𝐵,𝑣,Ψ+𝑎1𝑢,𝐵,𝑢,𝐵,𝑣,Ψ+𝑏𝑣,Ψ,𝑝𝑣=𝐹,Ψ.(5.8) Subtraction of (4.8) from (5.6) yields 𝑎0𝑢(𝑢,𝐵),𝐵,𝑣,Ψ+𝑎1𝑢(𝑢,𝐵),𝐵𝑣,(𝑢,𝐵),,Ψ+𝑎1𝑢,𝐵,𝑢(𝑢,𝐵),𝐵,𝑣,Ψ+𝑏𝑣,Ψ,𝑝𝑝𝑣=𝐸,Ψ.(5.9) Let (𝑤,Φ) be an arbitrary element of 𝑍×𝑋2, that is: 𝑏𝑤,Φ,𝜒=0,𝜒𝑀.(5.10) Then, 𝑏𝑢𝑤,𝐵Φ,𝜒=𝑏𝑢,𝐵,𝜒𝑏𝑤,Φ,𝜒=0.(5.11) For all (𝑣,Ψ)𝑊,𝑠𝑀, by virtue of (𝑢𝑤,𝐵Φ)𝑍×𝑋2 and (5.9), we get 𝑎0𝑤,Φ𝑢,𝐵,𝑣,Ψ+𝑎1𝑤,Φ𝑢,𝐵𝑣,(𝑢,𝐵),,Ψ+𝑎1𝑢,𝐵,𝑤,Φ𝑢,𝐵,𝑣,Ψ+𝑏𝑣,Ψ,𝑠𝑝=𝑎0𝑤,Φ𝑣(𝑢,𝐵),,Ψ+𝑎1𝑤,Φ𝑣(𝑢,𝐵),(𝑢,𝐵),,Ψ+𝑎1𝑢,𝐵,𝑤,Φ𝑣(𝑢,𝐵),,Ψ+𝑏𝑣,Ψ,𝑠𝑣𝑝+𝐸,Ψ.(5.12) Notice that 𝑎1𝑢,𝐵,𝑤,Φ𝑢,𝐵,𝑤,Φ𝑢,𝐵𝑏=0,𝑢𝑤,𝐵Φ,𝑠𝑝=0.(5.13) Let (𝑣,Ψ)=(𝑤,Φ)(𝑢,𝐵), and by (5.12), we obtain 𝑎0𝑤,Φ𝑢,𝐵,𝑤,Φ𝑢,𝐵+𝑎1𝑤,Φ𝑢,𝐵𝑤,(𝑢,𝐵),,Φ𝑢,𝐵=𝑎0𝑤,Φ𝑤(𝑢,𝐵),,Φ𝑢,𝐵+𝑎1𝑤,Φ𝑤(𝑢,𝐵),(𝑢,𝐵),,Φ𝑢,𝐵+𝑎1𝑢,𝐵,𝑤,Φ𝑤(𝑢,𝐵),,Φ𝑢,𝐵+𝑏𝑤,Φ𝑢,𝐵,𝑠𝑤𝑝+𝐸,Φ𝑢,𝐵.(5.14) Using the continuity properties of 𝑎0,𝑎1 and the stability bounds for (𝑢,𝐵)𝑊 and (𝑢,𝐵) in Theorems 2.1 and 4.6, respectively, the right-hand side of (5.14) can be bounded by r.h.s.(𝑤,Φ)(𝑢,𝐵)×𝐶𝛾2(𝑤,Φ)(𝑢,𝐵)+𝐶𝑐(𝑤,Φ)(𝑢,𝐵)(𝑢,𝐵)𝑊+𝐶𝑐(𝑤,Φ)(𝑢,𝐵)(𝑢,𝐵)𝑠+𝐶𝑝0+𝐸𝑤,Φ𝑢,𝐵(𝑤,Φ)(𝑢,𝐵)𝐶(𝑤,Φ)(𝑢,𝐵)(𝑤,Φ)(𝑢,𝐵)+𝑠𝑝0+𝐸𝑤,Φ𝑢,𝐵𝑤,Φ𝑢,𝐵.(5.15) Next, the coercivity property of the form 𝑎0, continuity of 𝑎1 in Lemma 4.4, stability bound for (𝑢,𝐵)𝑊 in Theorem 2.1, and the assumption 𝐶𝑐𝛾3𝑓1/𝐶𝑎𝐶1𝛾21<1/2 allow us to bound the left-hand side of (5.14) as l.h.s.𝐶𝑎𝛾1(𝑤,Φ)(𝑢,𝐵)2𝐶𝑐𝛾3(𝑤,Φ)(𝑢,𝐵)2(𝑢,𝐵)𝑊12𝐶𝑎𝛾1(𝑤,Φ)(𝑢,𝐵)2.(5.16) Combining these bounds, we have (𝑤,Φ)(𝑢,𝐵)𝑤𝐶,Φ(𝑢,𝐵)+𝑠𝑝0+𝐸𝑤,Φ𝑢,𝐵𝑤,Φ𝑢,𝐵.(5.17) Then, applying the triangle inequality, we get (𝑢,𝐵)(𝑢,𝐵)𝑤𝐶,Φ(𝑢,𝐵)+𝑠𝑝0+𝐸𝑤,Φ𝑢,𝐵𝑤,Φ𝑢,𝐵.(5.18) Now, for (𝑤,Φ)𝑍×𝑋2,𝑠𝑀, taking the infimum of (5.18) yields (𝑢,𝐵)(𝑢,𝐵)𝐶inf𝑤,Φ𝑍×𝑋2𝑤,Φ(𝑢,𝐵)+inf𝑠𝑀𝑠𝑝0+sup𝑣,Ψ𝑍×𝑋2𝐸𝑣,Ψ𝑣,Ψ.(5.19) With the argument as [37], we know that inf(𝑤,Φ)𝑍×𝑋2(𝑤,Φ)(𝑢,𝐵)𝐶inf(𝑣,Ψ)𝑊(𝑤,Φ)(𝑢,𝐵).(5.20) Substituting (5.20) into (5.19) implies (5.2).Step 2. For (𝑣,Ψ)𝑊,𝑠𝑀, we have from (5.9) that 𝑏𝑣,Ψ,𝑠𝑝=𝑏𝑣,Ψ,𝑠𝑝+𝑏𝑣,Ψ,𝑝𝑝=𝑏𝑣,Ψ,𝑠𝑝𝑎0𝑢(𝑢,𝐵),𝐵,𝑣,Ψ𝑎1𝑢(𝑢,𝐵),𝐵𝑣,(𝑢,𝐵),,Ψ𝑎1𝑢,𝐵𝑢,(𝑢,𝐵),𝐵,𝑣,Ψ𝑣+𝐸,Ψ.(5.21) Using the continuity properties of 𝑎0 and 𝑎1 and the discrete inf-sup condition (4.21) of Lemma 4.5, it follows that 𝑠𝑝01𝛽𝐶𝑠𝑝0+𝐶𝛾2+𝐶𝑐(𝑢,𝐵)𝑊+𝑢,𝐵𝑢(𝑢,𝐵),𝐵+𝐸𝑣,Ψ𝑣,Ψ.(5.22) Then, with the help of the triangle inequality and (5.2), we complete the proof.

Theorem 5.2. Let 𝑢(𝐻10(Ω)3𝐻2(Ω)3), 𝐵(𝐻2(Ω)3𝐻1𝑛(Ω)3), 𝑝(𝐿20(Ω)𝐻1(Ω)), and ((𝑢,𝐵),𝑝)𝑊×𝑀 be the solutions of Problems (𝐈1) and (𝐈2), respectively. Then there holds (𝑢,𝐵)(𝑢,𝐵)+𝑝𝑝0𝐶|𝑢|2+𝐵2+𝑝1.(5.23)

Proof. On the one hand, the interpolation theory gives inf𝑣𝑋1𝑢𝑣21𝑢Π1𝑢21𝐶2|𝑢|22,infΨ𝑋2𝐵Ψ21𝐶2𝐵22.(5.24) Therefore, by (5.24), we obtain inf(𝑣,Ψ)𝑊(𝑢,𝐵)(𝑣,Ψ)𝐶|𝑢|2+𝐵2.(5.25) At the same time, for 𝑝𝐿20(Ω), we define the interpolation 𝑅0𝑝𝑀 on each element 𝐾 as 𝐾𝑝𝑅0𝑝𝑑𝐱=0.(5.26) Then there holds inf𝑠𝑀𝑝𝑠0𝑝𝑅0𝑝0𝐶𝑝1.(5.27) On the other hand, by the similar techniques to [2527, 29, 30, 32], we have ||𝐸𝑣,Ψ||𝐶|𝑢|2+𝑝1(𝑣,Ψ).(5.28) Substituting (5.24)–(5.28) into (5.2) and (5.3) yields the desired result.
Next, we will establish the error estimates in 𝐿2-norm for the velocity and the magnetic fields by use of the duality argument introduced in [46].
We consider the following dual problem. Find (𝑤,Φ) and 𝑠 such that. 𝑀2Δ𝑤+𝑁1[]𝑤𝑢𝑢𝑤+𝑠+𝑅𝑚1(×Φ)×𝐵=𝑢𝑢𝑅,inΩ,𝑚2[]×(×Φ)(Φ)+𝑅𝑚1[](×𝐵)×𝑤(×Φ)×𝑢×(𝐵×𝑤)=𝐵𝐵,inΩ,𝑤=0,inΩ,𝑤=0,on𝜕Ω,𝐵𝑛=0,𝑅𝑚1(×Φ)×𝑛+𝑤×𝐵×𝑛=0,on𝜕Ω.(5.29)

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

Problem (𝐈3). Find (𝑤,Φ)𝑊(Ω) and 𝑠𝐿20(Ω) such that for all (𝑣,Ψ)𝑊(Ω),𝜓𝐿20(Ω)𝑎0((𝑣,Ψ),(𝑤,Φ))+𝑎1((𝑢,𝐵),(𝑣,Ψ),(𝑤,Φ))+𝑎1=𝑢((𝑣,Ψ),(𝑢,𝐵),(𝑤,Φ))+𝑏((𝑣,Ψ),𝑠)(𝑢,𝐵),𝐵,,(𝑣,Ψ)𝑏((𝑤,Φ),𝜓)=0.(5.30) Under the same hypotheses as Theorem 2.1, we may easily know that Problem (𝐈3) has a unique solution ((𝑤,Φ),𝑠)𝑊(Ω)×𝐿20(Ω).
We require that (5.29) be 𝐻2-regular, that is: (𝑤,Φ)2+𝑠1𝐶(𝑢,𝐵)(𝑢,𝐵)0.(5.31) Let ((𝑤,Φ),𝑠)𝑊×𝑀 satisfy (𝑤,Φ)(𝑤,Φ)+𝑠𝑠0𝐶|𝑤|2+Φ2+𝑠1.(5.32)

Theorem 5.3. Under the hypothesis of Theorem 5.2, let ((𝑤,Φ),𝑠) be the solution of Problem (𝐈3), and assume that (5.31) holds. Then we have (𝑢,𝐵)(𝑢,𝐵)0𝐶2|𝑢|2+𝐵2+𝑝1.(5.33)

Proof. By (5.31) and (5.32), we deduce that 𝑤(𝑤,Φ),Φ+𝑠𝑠0𝐶(𝑢,𝐵)(𝑢,𝐵)0.(5.34) Multiplying (𝑢𝑢) and (𝐵𝐵) both sides of the first and the second equation of (5.29), respectively, and integrating by parts on each element, we see that (𝑢,𝐵)(𝑢,𝐵)20=𝑎0𝑢(𝑢,𝐵),𝐵,(𝑤,Φ)+𝑎1𝑢(𝑢,𝐵),(𝑢,𝐵),𝐵,(𝑤,Φ)+𝑎1𝑢(𝑢,𝐵),𝐵,(𝑢,𝐵),(𝑤,Φ)+𝑏u(𝑢,𝐵),𝐵,𝑠(2𝑁)1𝐾Γ𝐾div𝑢𝑢(𝑢𝑤)𝑑𝐱+𝐹𝑢𝑢,(5.35) where 𝐹𝑢𝑢=𝑀2𝐾Γ𝜕𝐾𝜕𝑤𝜕𝑛𝑢𝑢𝑑𝑠(2𝑁)1𝐾Γ𝜕𝐾(𝑢𝑛)𝑤𝑢𝑢𝑑𝑠+(2𝑁)1𝐾Γ𝜕𝐾𝑢𝑢𝑛(𝑢𝑤)𝑑𝑠+𝐾Γ𝜕𝐾𝑠𝑢𝑢𝑛𝑑𝑠.(5.36) Subtraction of (5.9) yields 𝑎0𝑢(𝑢,𝐵),𝐵,𝑣,Ψ+𝑎1𝑢(𝑢,𝐵),𝐵𝑣,(𝑢,𝐵),,Ψ+𝑎1𝑢,𝐵,𝑢(𝑢,𝐵),𝐵,𝑣,Ψ+𝑏𝑣,Ψ,𝑝𝑝𝑣=𝐸,Ψ.(5.37) Note that 𝑏𝑢(𝑢,𝐵),𝐵,𝜙=0,𝜙𝑀.(5.38) Now, setting 𝜓=𝑝𝑝 in Problem (𝐈𝟑), we have 𝑏(𝑤,Φ),𝑝𝑝=0.(5.39) From (5.35)–(5.39), we get 𝑢(𝑢,𝐵),𝐵20=𝑎0𝑢(𝑢,𝐵),𝐵,𝑣(𝑤,Φ),Ψ+𝑏𝑢(𝑢,𝐵),𝐵,𝑠𝜙+𝑏𝑣(𝑤,Φ),Ψ,𝑝𝑝+𝐴1+𝐴2+𝐴3,(5.40) where 𝐴1=𝑎1𝑢,𝐵𝑢,(𝑢,𝐵),B𝑣,(𝑤,Φ),Ψ+𝑎1𝑢(𝑢,𝐵),𝐵,𝑣(𝑢,𝐵),(𝑤,Φ),Ψ+𝑎1𝑢(𝑢,𝐵),𝐵𝑢,(𝑢,𝐵),𝐵,𝐴,(𝑤,Φ)2=𝐹𝑢𝑢𝑣+𝐸,Ψ,𝐴31=2𝑁𝐾Γ𝐾div𝑢𝑢(𝑢𝑤)𝑑𝐱(5.41) By (2.8), Lemma 4.4, and Theorem 4.6, we find 𝐴1𝑢𝐶(𝑢,𝐵),𝐵𝑣(𝑤,Φ),Ψ+𝑢(𝑢,𝐵),𝐵2(𝑤,Φ)𝑊.(5.42) From [45], we know 𝐹𝑢𝑢𝐶|𝑤|2+𝑠1𝑢𝑢1.(5.43) By virtue of 𝑢,𝑤𝐻2(Ω)3𝐶0(Ω)2, we obtain 𝐸𝑣,Ψ=𝐾Γ𝜕𝐾𝑀2𝜕𝑢𝑣𝜕𝑛𝑣𝑤𝑝𝑤𝑛(2𝑁)1𝑣(𝑢𝑛)𝑢𝑤𝑑𝑠𝐶|𝑢|2+𝑝1𝑣𝑤1.(5.44) Let 𝑎𝐾 be a constant such that 𝑢𝑤𝑎𝐾0,𝐾𝐶𝑢𝑤1,𝐾𝐶𝑢1,𝐾𝑤2,𝐾.(5.45) Since div𝑢=0,𝑏((𝑢,𝐵),𝑞)=0, 𝑞𝑀 and (5.45), we obtain ||𝐴3||=|||||12𝑁𝐾Γ𝐾div𝑢𝑢𝑢𝑤𝑎𝐾|||||𝑑𝐱𝐶𝑢𝑢1𝑢1𝑤2.(5.46) Thus, by (5.31) and the approximation theory, there hold inf(𝑣,Ψ)𝑊(𝑤,Φ)(𝑣,Ψ)𝐶(𝑤,Φ)2𝐶(𝑢,𝐵)(𝑢,𝐵)0,inf𝑣𝑋2𝑤𝑣1𝐶𝑤2𝐶(𝑢,𝐵)(𝑢,𝐵)0,inf𝜙𝑀𝑠𝜙0𝐶𝑠1𝐶(𝑢,𝐵)(𝑢,𝐵)0,(𝑤,Φ)𝑊(𝑤,Φ)2𝐶(𝑢,𝐵)(𝑢,𝐵)0,||𝑏𝑢(𝑢,𝐵),𝐵,𝑠𝜙||𝑢𝐶(𝑢,𝐵),𝐵𝑠𝜙0,||𝑏𝑣(𝑤,Φ),Ψ,𝑝𝑝||𝑢𝐶(𝑢,𝐵),𝐵𝑠𝜙0.(5.47) 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 𝑢=0,𝑛×𝐵=0,(×𝐵)𝑛=0 on 𝜕Ω when 𝑢𝐻10(Ω)3,𝐵𝐻={𝐵𝐻1(Ω)3;(𝐵×𝑛)|𝜕Ω=0}.

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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