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

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 [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 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 𝐶^1,1, 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 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 [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 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, [28–30] obtained maximum norm estimates of 𝑃₁/𝑃₀ element and the optimal error estimates of 𝐸𝑄^1rot/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, 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 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 ℎ, 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 −𝑀⁻²Δ𝑅𝑢+𝑁⁻¹𝑢⋅∇𝑢+∇𝑝−𝑅^𝑚−1(∇×𝐵)×𝐵=𝑓inΩ,_𝑚−1∇×((∇∇××𝐵∇∇𝐵𝑢𝐵⋅⋅𝑛)=⋅)𝐵𝑢×−=0==𝑛∇0=00o×n0o(iinn𝑢n𝜕Ω×oΩ𝜕ΩnΩ,,𝐵,,𝜕)Ω=,0inΩ,(2.1) where Ω is a simply connected, bounded domain with unit outward normal 𝑛=(𝑛₁,𝑛₂,𝑛₃) on 𝜕Ω. 𝑀, 𝑁, and 𝑅_𝑚 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. 𝑓∈𝐻⁻¹(Ω)³ is the body force.
Set 𝐿𝐻¹⁰(Ω)³=𝑣∈𝐻¹(Ω)³;𝑣|_𝜕Ω=0₂₀,𝐻(Ω)=𝑞∈𝐿²(Ω);_Ω𝑞𝑑𝐱=0_1𝑛,(Ω)³=𝑣∈𝐻¹(Ω)³;(𝑣⋅𝑛)|_𝜕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), 𝑝∈𝐿²⁰(Ω) such that 𝑎((𝑢,𝐵),(𝑢,𝐵),(𝑣,𝑏Ψ()()𝑢+,𝐵𝑏)(,(𝜒𝑣,)Ψ=),0𝑝,)=∀𝜒𝐹∈((𝑣𝐿,₂₀Ψ)),∀(𝑣,Ψ)∈𝑊(Ω),(Ω),(2.3) where 𝑎((𝑢,𝐵),(𝑣,Ψ),(𝑤,Φ))𝑊∶=(Ω𝑎)₀=𝐻¹⁰(Ω)³×𝐻^1𝑛(Ω)³,𝑎((𝑣,Ψ),(𝑤,Φ))+𝑎₁((𝑢,𝐵),(𝑣,Ψ),(𝑤,Φ)),₀((𝑣,Ψ),(𝑤,Φ))∶=𝑀⁻²_Ω∇𝑣∶∇𝑤𝑑𝐱+𝑅^𝑚−2_Ω𝑎[(∇×Ψ)⋅(∇×Φ)+(∇⋅Ψ)(∇⋅Φ)]𝑑𝐱,₁((𝑢,𝐵𝑐),(𝑣,Ψ),(𝑤,Φ))∶=𝑐₀(𝑢;𝑣,𝑤)−𝑐₁(𝐵;𝑤,Ψ)+𝑐₂(𝐵;𝑣,Φ),₀(𝑢;𝑐𝑣,𝑤)∶=_Ω(2𝑁)⁻¹(𝑢⋅∇𝑣⋅𝑤−𝑢⋅∇𝑤⋅𝑣)𝑑𝐱,₁𝑐(𝐵;𝑤,Ψ)∶=_Ω𝑅^𝑚−1(∇×Ψ)×𝐵⋅𝑤𝑑𝐱,₂𝑏((𝑣,Ψ),𝜒()𝐵∶;=𝑣,−Φ)_Ω∶=_Ω𝑅^𝑚−1(∇×Φ)×𝐵⋅𝑣𝑑𝐱,𝜒∇⋅𝑣𝑑𝐱,𝐹((𝑣,Ψ))∶=_Ω𝑓𝑣𝑑𝐱.(2.4)
It has been shown in [9, 37, 38] that for 𝑢,𝑣,𝑤∈𝐻¹⁰(Ω)³, 𝐵,Ψ,Φ∈𝐻^1𝑛(Ω)³, there hold 𝑎𝑐₀(𝑢;𝑣,𝑤)=−𝑐₀(𝑢;𝑤,𝑣),𝑐₀(𝑤;𝑣,𝑣)=0,₁((𝑢,𝐵),(𝑣,𝑎Ψ),(𝑤,Φ))=−𝑎₁((𝑢,𝐵),(𝑤,Φ),(𝑣,Ψ)),₁((𝑢,𝐵),(𝑣,Ψ),(𝑣,Ψ))=0.(2.5)
Let 𝑍(Ω)={𝑣∈𝐻¹⁰(Ω)³,∇⋅𝑣=0}. For 𝑣∈𝐻¹⁰(Ω)³ and Ψ∈𝐻^1𝑛(Ω)³, we will equip 𝑊(Ω) with the norm ‖𝑓‖(‖𝑣,₋₁Ψ)‖_𝑊∶=‖𝑣‖²¹+‖Ψ‖²¹^1/2,∶=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 𝑓∈𝐻⁻¹(Ω)³, then Problem (𝐈₁) has at least a solution, in addition, that is unique provided that 𝐶₂𝛾₃𝐶₁𝛾₁⁻²‖𝑓‖₋₁<1(2.7) and satisfying the stability bound ‖(𝑢,𝐵)‖_𝑊≤𝐶₁𝛾₁⁻¹‖𝑓‖₋₁,(2.8) where 𝛾₁=min{𝑀⁻²,𝑅^𝑚−2}, 𝛾₂=max{𝑀⁻²,𝑅^𝑚−2}, 𝛾₃=max{𝑁⁻¹,𝑅^𝑚−1} and 𝐶₁,𝐶₂ are positive constants only depending on the domain Ω.

3. Nonconforming Mixed Finite Element Schemes

Let Γ^ℎ={𝐾} be regular and quasi-uniform tetrahedra or hexahedra partition of Ω with mesh size ℎ. We use the finite element spaces 𝑋_1ℎ⊈𝐻¹⁰(Ω)³, 𝑀_ℎ⊂𝐿²⁰(Ω) and 𝑋_2ℎ⊂𝐻^1𝑛(Ω)³ to approximate the unknown variables 𝑢, 𝑝, and 𝐵. The following assumptions about the space pair (𝑋_1ℎ,𝑀_ℎ) are provided:(A) for all 𝐾∈Γ^ℎ, 𝑃₁(𝐾)³⊂𝑋_1ℎ;(B)𝑀_ℎ={𝜒^ℎ∈𝐿²⁰(Ω);𝜒^ℎ|_𝐾 a constant, ∀𝐾∈Γ^ℎ};(C)‖⋅‖_1ℎ=(∑_𝐾∈Γℎ|⋅|^21,𝐾)^1/2 is a norm 𝑋_1ℎ;(D)for all 𝑣^ℎ∈𝑋_1ℎ,∫_𝐹[𝑣^ℎ]𝑑𝑠=0, 𝐹⊂𝜕𝐾;(E)for all 𝑣∈𝐻¹⁰(Ω)³, 𝑞^ℎ∈𝑀_ℎ, 𝑏_1ℎ(𝑣−Π¹𝑣,𝑞^ℎ)=0, ‖Π¹𝑣‖_1ℎ≤𝐶|𝑣|₁, where [𝑣^ℎ] stands for the jump of 𝑣^ℎ across the face 𝐹 if 𝐹 is an internal face, and it is equal to 𝑣^ℎ itself if 𝐹⊂𝜕Ω, Π¹ is the interpolation operator associated with 𝑋_1ℎ satisfying Π_𝐾=Π¹|_𝐾 for 𝐾∈Γ^ℎ, and 𝑃₁(𝐾) is the polynomial space of degree less than or equal to one on 𝐾.

Introduce the finite element space 𝑅₁(𝐾)=𝑃¹𝑄(𝐾)if𝐾istetrahedra,₁(𝐾)if𝐾ishexahedra.(3.1) The finite element space 𝑋_2ℎ is defined by 𝑋_2ℎ=Ψ^ℎ∈𝐻^1𝑛(Ω)³;𝑞^ℎ||_𝐾∈𝑅₁(𝐾)³,Ψ^ℎ⋅𝑛||_𝜕Ω=0,∀𝐾∈Γ^ℎ,(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 [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 𝑊_ℎ=𝑋_1ℎ×𝑋_2ℎ and the trilinear forms 𝑎_ℎ,𝑎_1ℎ,𝑐_𝑖ℎ(𝑖=0,1,2) and the bilinear forms 𝑎_0ℎ and 𝑏_ℎ be defined as follows:

for (𝑢^ℎ,𝐵^ℎ),(𝑣^ℎ,Ψ^ℎ),(𝑤^ℎ,Φ^ℎ)∈𝑊_ℎ and 𝜒^ℎ∈𝑀_ℎ,𝑎_ℎ∶𝑢=^ℎ,𝑎𝐵^ℎ_0ℎ,𝑣^ℎ,Ψ^ℎ,𝑤^ℎ,Φ^ℎ𝑣^ℎ,Ψ^ℎ,𝑤^ℎ,Φ^ℎ+𝑎_1ℎ𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ,𝑤^ℎ,Φ^ℎ,(4.1)𝑎_0ℎ=𝑣^ℎ,Ψ^ℎ,𝑤^ℎ,Φ^ℎ_𝐾∈Γℎ𝑀⁻²_𝐾∇𝑣^ℎ⋅∇𝑤^ℎ+𝑅^𝑚−2_𝐾∇×Ψ^ℎ⋅∇×Φ^ℎ+∇⋅Ψ^ℎ∇⋅Φ^ℎ𝑑𝐱,(4.2)𝑎_1ℎ𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ,𝑤^ℎ,Φ^ℎ∶=𝑐_0ℎ𝑢^ℎ;𝑣^ℎ,𝑤^ℎ−𝑐_1ℎ𝐵^ℎ;𝑤^ℎ,Ψ^ℎ+𝑐_2ℎ𝐵^ℎ;𝑣^ℎ,Φ^ℎ,(4.3)𝑐_0ℎ𝑢^ℎ;𝑣^ℎ,𝑤^ℎ∶=_𝐾∈Γℎ_𝐾(2𝑁)⁻¹𝑢^ℎ⋅∇𝑣^ℎ⋅𝑤^ℎ−𝑢^ℎ⋅∇𝑤^ℎ⋅𝑣^ℎ𝑑𝐱,(4.4)𝑐_1ℎ𝐵^ℎ;𝑤^ℎ,Ψ^ℎ=∶_𝐾∈Γℎ_𝐾𝑅^𝑚−1∇×Ψ^ℎ×𝐵^ℎ⋅𝑤^ℎ𝑑𝐱,(4.5)𝑐_2ℎ𝐵^ℎ;𝑣^ℎ,Φ^ℎ∶=_𝐾∈Γℎ_𝐾𝑅^𝑚−1∇×Φ^ℎ×𝐵^ℎ⋅𝑣^ℎ𝑑𝐱(4.6)𝑏_ℎ𝑣^ℎ,Ψ^ℎ,𝜒^ℎ∶=−_𝐾∈Γℎ_𝐾𝜒^ℎ∇⋅𝑣^ℎ𝑑𝐱,(4.7) respectively.

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

Problem (𝐈₂). Find (𝑢^ℎ,𝐵^ℎ)∈𝑊_ℎ, 𝑝^ℎ∈𝑀_ℎ such that for all (𝑣^ℎ,Ψ^ℎ)∈𝑊_ℎ, 𝜒^ℎ∈𝑀_ℎ, 𝑎_ℎ𝑢^ℎ,𝐵^ℎ,𝑢^ℎ,𝐵^ℎ,𝑏𝑣^ℎ,Ψ^ℎ+𝑏_ℎ𝑣^ℎ,Ψ^ℎ,𝑝^ℎ=𝐹𝑣^ℎ,Ψ^ℎ_ℎ,𝑢^ℎ,𝐵^ℎ,𝜒^ℎ=0.(4.8)
From the definition of (4.3), 𝑎_1ℎ satisfies the following antisymmetric properties [9]: 𝑎𝑎_1ℎ𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ,𝑣^ℎ,Ψ^ℎ=0,_1ℎ𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ,𝑤^ℎ,Φ^ℎ=−𝑎_1ℎ𝑢^ℎ,𝐵^ℎ,𝑤^ℎ,Φ^ℎ,𝑣^ℎ,Ψ^ℎ.(4.9)
Let 𝑍_ℎ={𝑣^ℎ∈𝑋_1ℎ,𝑏((𝑣^ℎ,Ψ^ℎ),𝜒^ℎ)=0}. For all 𝑣^ℎ=(𝑣^ℎ1,𝑣^ℎ2,𝑣^ℎ3)∈𝑋_1ℎ,Ψ^ℎ=(Ψ^ℎ1,Ψ^ℎ2,Ψ^ℎ3)∈𝑋_2ℎ, we define ‖‖𝑣^ℎ‖‖_0ℎ=⎛⎜⎜⎝_𝐾∈Γℎ‖‖𝑣^ℎ‖‖^20,𝐾⎞⎟⎟⎠^1/2,‖‖𝑣^ℎ‖‖_1ℎ=⎛⎜⎜⎝_𝐾∈Γℎ||𝑣^ℎ||^21,𝐾⎞⎟⎟⎠^1/2‖‖(𝑣_ℎ,,Ψ^ℎ)‖‖_ℎ=⎛⎜⎜⎝_𝐾∈Γℎ‖‖𝑣^ℎ‖‖^21ℎ+‖‖Ψ^ℎ‖‖²¹⎞⎟⎟⎠^1/2‖𝑓‖_ℎ,∶=sup_{(0,0)≠(𝑣ℎ,Ψℎ)∈𝑋1ℎ×𝑋2ℎ}𝐹𝑣^ℎ,Ψ^ℎ‖‖(𝑣^ℎ,Ψ^ℎ)‖‖_ℎ,(4.10) respectively. Then it is easy to see that ‖⋅‖_0ℎ and ‖⋅‖_1ℎ are the norms over 𝑋_1ℎ and ‖(⋅,⋅)‖_ℎ is the norm over 𝑊_ℎ.

Lemma 4.1. The following discrete Poincaré -Friedrichs inequality holds: ‖‖Ψ^ℎ‖‖₀≤𝐶‖‖∇×Ψ^ℎ‖‖₀,∀Ψ^ℎ∈𝑋_2ℎ.(4.11)

Proof. We consider the following problem: ∇∇××𝐵∇𝐵⋅∇⋅𝑛𝐵×=×=𝐵𝑛00==o0inn𝑓𝜕ΩoΩni,n,𝜕ΩΩ,.(4.12) Then by [3], the solution 𝐵 of (4.12) satisfies ‖‖∇×𝐵‖‖₀≤𝐶‖𝑓‖₀.(4.13) On the one hand, by Green's formula and Hölder's inequality, we deduce that ||||_Ω𝑓Ψ^ℎ𝑑𝐱||||=|||||_𝐾∈Γℎ≤‖‖∇×𝐵_𝐾‖‖₀∇×𝐵∇×Ψ^ℎ𝑑𝐱|||||‖‖∇×Ψ^ℎ‖‖₀.(4.14) Using (4.13)-(4.14) and choosing 𝑓=Ψ^ℎ, 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^ℎ,𝐵^ℎ), (𝑣^ℎ,Ψ^ℎ), and (𝑤^ℎ,Φ^ℎ)∈𝑊_ℎ, we have(1)|𝑐_0ℎ(𝑢^ℎ;𝑣^ℎ,𝑤^ℎ)|≤𝐶‖𝑢^ℎ‖_1ℎ‖𝑣^ℎ‖_1ℎ‖𝑤^ℎ‖_1ℎ, (2)|𝑐_1ℎ(𝐵^ℎ;𝑤^ℎ,Ψ^ℎ)|≤𝐶‖∇×Ψ^ℎ‖₀‖𝐵^ℎ‖₁‖𝑤^ℎ‖_1ℎ,(3)|𝑐_2ℎ(𝐵^ℎ;𝑣^ℎ,Φ^ℎ)|≤𝐶‖∇×Φ^ℎ‖₀‖𝐵^ℎ‖₁‖𝑣^ℎ‖_1ℎ.

Proof. The first result is wellknown [30, 37, 38]. To prove the second result, we need the imbedding properties 𝐻^1𝑛(Ω)³↪𝐻¹(Ω)³↪𝐿⁴(Ω)³ and the discrete imbedding inequality showed in [30, 32]: ‖‖𝑣^ℎ‖‖_0,2𝑘,Ω≤𝐶‖‖𝑣^ℎ‖‖_1ℎ,∀𝑣^ℎ∈𝑋_1ℎ,𝑘=1,2.(4.15) Thus, ||𝑐_1ℎ𝐵^ℎ;𝑤^ℎ,Ψ^ℎ||≤_𝐾∈Γℎ_𝐾1𝑅_𝑚≤1||∇×Ψ^ℎ×𝐵^ℎ⋅𝑤^ℎ||𝑑𝐱𝑅_𝑚‖‖(∇×Ψ^ℎ)‖‖₀‖‖𝐵^ℎ‖‖_0,4‖‖𝑤^ℎ‖‖_0,4≤𝐶‖‖(∇×Ψ^ℎ)‖‖₀‖‖𝐵^ℎ‖‖₁‖‖𝑤^ℎ‖‖_1ℎ,(4.16) the assertion for 𝑐_1ℎ is proved. The proof for 𝑐_2ℎ is analogous.

Lemma 4.4. Let (𝑢^ℎ,𝐵^ℎ), (𝑣^ℎ,Ψ^ℎ), and (𝑤^ℎ,Φ^ℎ)∈𝑊_ℎ; then the following results hold: (1)|𝑎_1ℎ((𝑢^ℎ,𝐵^ℎ),(𝑣^ℎ,Ψ^ℎ),(𝑤^ℎ,Φ^ℎ))|≤𝐶_𝑐𝛾₃‖(𝑢^ℎ,𝐵^ℎ)‖_ℎ‖(𝑣^ℎ,Ψ^ℎ)‖_ℎ‖(𝑤^ℎ,Φ^ℎ)‖_ℎ, (2)|𝑎_0ℎ((𝑢^ℎ,𝐵^ℎ),(𝑢^ℎ,𝐵^ℎ))|≥𝐶_𝑎𝛾₁‖(𝑢^ℎ,𝐵^ℎ)‖^2ℎ, (3)|𝑎_0ℎ((𝑢^ℎ,𝐵^ℎ),(𝑣^ℎ,Ψ^ℎ))|≤𝐶𝛾₂‖(𝑢^ℎ,𝐵^ℎ)‖_ℎ‖(𝑣^ℎ,Ψ^ℎ)‖_ℎ,where 𝐶_𝑐,𝐶_𝑎 are positive constants, independent of ℎ.

Proof. Firstly, using the triangle inequality and Lemma 4.3 yields ||𝑎_1ℎ≤𝑢||𝑐^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ,𝑤^ℎ,Φ^ℎ_0ℎ||≤𝐶_𝑐𝑢^ℎ;𝑣^ℎ,𝑤^ℎ||+||𝑐_1ℎ𝐵^ℎ;𝑤^ℎ,Ψ^ℎ||+||𝑐_2ℎ𝐵^ℎ;𝑣^ℎ,Φ^ℎ||𝛾₃‖‖(𝑢^ℎ,𝐵^ℎ)‖‖_ℎ‖‖(𝑣^ℎ,Ψ^ℎ)‖‖_ℎ‖‖(𝑤^ℎ,Φ^ℎ)‖‖_ℎ.(4.17) Applying 𝐻^1𝑛(Ω)³↪𝐻¹(Ω)³ and the following inequality [9, 37, 38] ‖𝑣‖₀≤𝐶‖∇×𝑣‖₀+‖∇⋅𝑣‖₀,∀𝑣∈𝐻^1𝑛(Ω)³(4.18) leads to 𝑎_0ℎ=𝑢^ℎ,𝐵^ℎ,𝑢^ℎ,𝐵^ℎ_𝐾∈Γℎ𝑀⁻²_𝐾∇𝑢^ℎ⋅∇𝑢^ℎ+𝑅^𝑚−2=𝑀₋₂_𝐾∇×𝐵^ℎ⋅∇×𝐵^ℎ+∇⋅𝐵^ℎ∇⋅𝐵^ℎ𝑑𝐱‖‖∇𝑢^ℎ‖‖^20ℎ+𝑅^𝑚−2‖‖∇×𝐵^ℎ‖‖²⁰+‖‖∇⋅𝐵^ℎ‖‖²⁰≥𝐶_𝑎=𝐶_𝑎min𝑀⁻²,𝑅^𝑚−2‖‖𝑢^ℎ‖‖^21ℎ+‖‖𝐵^ℎ‖‖²¹𝛾₁‖‖(𝑢^ℎ,𝐵^ℎ)‖‖^2ℎ.(4.19) With the help of Hölder's inequality, we find ||𝑎_0ℎ≤𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ_𝐾∈Γℎ||𝑀⁻²_𝐾||∇𝑢^ℎ⋅∇𝑣^ℎ||+𝑅^𝑚−2≤𝑀_𝐾||∇×𝐵^ℎ⋅∇×Ψ^ℎ||+||∇⋅𝐵^ℎ∇⋅Ψ^ℎ||𝑑𝐱₋₂≤𝐶max‖‖∇𝑀𝑢^ℎ‖‖_0ℎ−2‖‖∇𝑣^ℎ‖‖_0ℎ+𝐶𝑅^𝑚−2‖‖∇×𝐵^ℎ‖‖₀‖‖∇×Ψ^ℎ‖‖₀+‖‖∇⋅𝐵^ℎ‖‖₀‖‖∇⋅Ψ^ℎ‖‖₀,𝑅^𝑚−2‖‖(𝑢^ℎ,𝐵^ℎ)‖‖_ℎ‖‖(𝑣^ℎ,Ψ^ℎ)‖‖_ℎ=𝐶𝛾₂‖‖(𝑢^ℎ,𝐵^ℎ)‖‖_ℎ‖‖(𝑣^ℎ,Ψ^ℎ)‖‖_ℎ.(4.20) The proof is completed.

Lemma 4.5. The spaces 𝑋_1ℎ and 𝑀_ℎ satisfy the discrete inf-sup condition [37, 38]; that is, there exists 𝛽^⋆>0 such that inf_{𝜒ℎ∈𝑀ℎ}sup_{(𝑣ℎ,Ψℎ)∈𝑊ℎ}𝑏_ℎ𝑣^ℎ,Ψ^ℎ,𝜒^ℎ‖‖(𝑣^ℎ,Ψ^ℎ)‖‖_ℎ‖‖𝜒^ℎ‖‖₀≥𝛽^⋆.(4.21)

Proof. On the one hand, by [37, 38], there exists a constant 𝛽>0 such that inf_{𝜒∈𝐿20(Ω)}sup_{(𝑣,Ψ)∈𝑊(Ω)}𝑏((𝑣,Ψ),𝜒)‖(𝑣,Ψ)‖_𝑊‖𝜒‖₀≥𝛽.(4.22) Therefore, by the assumption (E) and (4.22), we obtain sup_{(𝑣ℎ,Ψℎ)∈𝑊ℎ}𝑏_ℎ𝑣^ℎ,Ψ^ℎ,𝜒^ℎ‖‖(𝑣^ℎ,Ψ^ℎ)‖‖_ℎ≥sup_{(𝑣,Ψℎ)∈𝐻10(Ω)3×𝑋2ℎ}𝑏_ℎΠ¹𝑣,Ψ^ℎ,𝜒^ℎ‖‖(Π¹𝑣,Ψ^ℎ)‖‖_ℎ=sup_{(𝑣,Ψℎ)∈𝐻10(Ω)3×𝑋2ℎ}𝑏_ℎ𝑣,Ψ^ℎ,𝜒^ℎ‖‖(Π¹𝑣,Ψ^ℎ)‖‖_ℎ≥1𝐶sup_{(𝑣,Ψℎ)∈𝐻10(Ω)3×𝑋2ℎ}𝑏𝑣,Ψ^ℎ,𝜒^ℎ‖‖(𝑣,Ψ^ℎ)‖‖_ℎ≥𝛽^∗‖‖𝜒^ℎ‖‖₀,(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 𝑓∈𝐻⁻¹(Ω)³, Problem (𝐈₂) has at least one solution ((𝑢^ℎ,𝐵^ℎ),𝑝^ℎ)∈𝑊_ℎ×𝑀_ℎ satisfying the stability bound ‖(𝑢^ℎ,𝐵^ℎ)‖_ℎ≤(𝐶_𝑎𝛾₁)⁻¹‖f‖_ℎ. Moreover, Problem (𝐈₂) has a unique solution provided that 𝐶_𝑐𝛾₃(𝐶_𝑎𝛾₁)⁻²‖𝑓‖_ℎ<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 𝐶_𝑐𝛾₃‖𝑓‖₋₁𝐶_𝑎𝐶₁𝛾²¹<12.(5.1) Let ((𝑢,𝐵),𝑝)∈𝑊(Ω)×𝐿²⁰(Ω) and ((𝑢^ℎ,𝐵^ℎ),𝑝^ℎ)∈𝑊_ℎ×𝑀_ℎ be the solutions of Problems (𝐈₁) and (𝐈₂), respectively. Then there hold (1)‖‖(𝑢,𝐵)−(𝑢^ℎ,𝐵^ℎ)‖‖_ℎ≤𝐶⎧⎪⎨⎪⎩inf_{𝑣ℎ,Ψℎ∈𝑊ℎ}‖‖(𝑢,𝐵)−𝑣^ℎ,Ψ^ℎ‖‖_ℎ+inf_{𝑠ℎ∈𝑀ℎ}‖‖𝑝−𝑠^ℎ‖‖₀+sup_{𝑣ℎ,Ψℎ∈𝑍ℎ×𝑋2ℎ}||𝐸𝑣^ℎ,Ψ^ℎ||‖‖𝑣^ℎ,Ψ^ℎ‖‖_ℎ⎫⎪⎬⎪⎭,(5.2)(2)‖‖𝑝−𝑝^ℎ‖‖₀≤𝐶⎧⎪⎨⎪⎩inf_{𝑣ℎ,Ψℎ∈𝑊ℎ}‖‖(𝑢,𝐵)−𝑣^ℎ,Ψ^ℎ‖‖_ℎ+inf_{𝑠ℎ∈𝑀ℎ}‖‖𝑝−𝑠^ℎ‖‖₀+sup_{𝑣ℎ,Ψℎ∈𝑊ℎ}||𝐸𝑣^ℎ,Ψ^ℎ||‖‖𝑣^ℎ,Ψ^ℎ‖‖_ℎ⎫⎪⎬⎪⎭,(5.3) where 𝐸𝑣^ℎ,Ψ^ℎ=_𝐾∈Γℎ_𝜕𝐾𝑀⁻²𝜕𝑢𝜕𝑛𝑣^ℎ−𝑝𝑣^ℎ⋅𝑛−(2𝑁)⁻¹(𝑢⋅𝑛)𝑢⋅𝑣^ℎ𝑑𝑠.(5.4)

Proof. We proceed in two steps.Step 1. For (𝑣^ℎ,Ψ^ℎ)∈𝑊_ℎ, by Green's formula, we have 𝑎_0ℎ=(𝑢,𝐵),𝑣^ℎ,Ψ^ℎ+𝑎_1ℎ(𝑢,𝐵),(𝑢,𝐵),𝑣^ℎ,Ψ^ℎ+𝑏_ℎ𝑣^ℎ,Ψ^ℎ,𝑝−𝐹𝑣^ℎ,Ψ^ℎ_𝐾∈Γℎ_𝐾𝑀⁻²∇𝑢⋅∇𝑣^ℎ𝑑𝐱+𝑅^𝑚−2_𝐾+(2𝑁)₋₁(∇×𝐵)⋅∇×Ψ^ℎ𝑑𝐱_𝐾𝑢⋅∇𝑢⋅𝑣^ℎ−𝑅_𝑚−1−𝑢⋅∇𝑣^ℎ⋅𝑢𝑑𝐱_𝐾(∇×𝐵)×𝐵⋅𝑣^ℎ−−∇×Ψ^ℎ×𝐵⋅𝑢𝑑𝐱_𝐾=𝑝⋅∇𝑣^ℎ𝑑𝐱−_𝐾𝑓𝑣^ℎ𝑑𝐱_𝐾∈Γℎ_𝐾−𝑀⁻²Δ𝑢⋅𝑣^ℎ𝑑𝐱+_𝜕𝐾𝑀⁻²𝜕𝑢𝜕𝑛𝑣^ℎ+𝑑𝑠_𝐾+𝑅^𝑚−2∇×(∇×𝐵)⋅Ψ^ℎ𝑑𝐱+_𝜕𝐾𝑅^𝑚−2(∇×𝐵×𝑛)⋅Ψ^ℎ𝑑𝑠_𝐾−𝑅_𝑚−1𝑁⁻¹𝑢⋅∇𝑢⋅𝑣^ℎ𝑑𝐱−_𝜕𝐾(2𝑁)⁻¹(𝑢⋅𝑛)𝑢⋅𝑣^ℎ𝑑𝑠_𝐾(∇×𝐵)×𝐵⋅𝑣^ℎ𝑑𝐱−𝑅^𝑚−1_𝐾+(∇×𝑢×𝐵)⋅Ψ^ℎ𝑑𝐱_𝜕𝐾=(𝑢×𝐵×𝑛)⋅Ψ^ℎ𝑑𝑠+_𝐾∇𝑝⋅𝑣^ℎ𝑑𝐱−_𝜕𝐾𝑝𝑣^ℎ⋅𝑛𝑑𝑠−_𝐾𝑓𝑣^ℎ𝑑𝐱_𝐾∈Γℎ_𝐾−𝑀⁻²Δ𝑢+𝑁⁻¹𝑢⋅∇𝑢+∇𝑝−𝑅^𝑚−1(∇×𝐵)×𝐵−𝑓⋅𝑣^ℎ+𝑅^𝑚−1_𝐾𝑅^𝑚−1+∇×(∇×𝐵)−∇×(𝑢×𝐵)⋅Ψ^ℎ𝑑𝐱_𝜕𝐾𝑀⁻²𝜕𝑢𝜕𝑛𝑣^ℎ=𝐸𝑣_ℎ−(2𝑁)⁻¹(𝑢⋅𝑛)𝑢⋅𝑣^ℎ−𝑝𝑣^ℎ⋅𝑛𝑑𝑠,Ψ^ℎ.(5.5) Thus, 𝑎_0ℎ=(𝑢,𝐹𝐵),𝑣_ℎ𝑣^ℎ,Ψ^ℎ+𝑎_1ℎ(𝑢,𝐵),(𝑢,𝐵),𝑣^ℎ,Ψ^ℎ+𝑏_ℎ𝑣^ℎ,Ψ^ℎ,𝑝,Ψ^ℎ+𝐸𝑣^ℎ,Ψ^ℎ.(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) 𝑎_0ℎ=𝑢𝐹^ℎ,𝐵^ℎ𝑣_ℎ,𝑣^ℎ,Ψ^ℎ+𝑎_1ℎ𝑢^ℎ,𝐵^ℎ,𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ+𝑏_ℎ𝑣^ℎ,Ψ^ℎ,𝑝^ℎ,Ψ^ℎ.(5.8) Subtraction of (4.8) from (5.6) yields 𝑎_0ℎ+(𝑢𝑎,𝐵_1ℎ)−𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ+𝑎_1ℎ(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,(𝑢,𝐵),𝑣^ℎ,Ψ^ℎ+𝑏_ℎ𝑢^ℎ,𝐵^ℎ,(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ𝑣^ℎ,Ψ^ℎ,𝑝−𝑝^ℎ=𝐸𝑣^ℎ,Ψ^ℎ.(5.9) Let (𝑤^ℎ,Φ^ℎ) be an arbitrary element of 𝑍_ℎ×𝑋_2ℎ, that is: 𝑏_ℎ𝑤^ℎ,Φ^ℎ,𝜒^ℎ=0,∀𝜒^ℎ∈𝑀_ℎ.(5.10) Then, 𝑏_ℎ𝑢^ℎ−𝑤^ℎ,𝐵^ℎ−Φ^ℎ,𝜒^ℎ=𝑏_ℎ𝑢^ℎ,𝐵^ℎ,𝜒^ℎ−𝑏_ℎ𝑤^ℎ,Φ^ℎ,𝜒^ℎ=0.(5.11) For all (𝑣^ℎ,Ψ^ℎ)∈𝑊_ℎ,𝑠^ℎ∈𝑀_ℎ, by virtue of (𝑢^ℎ−𝑤^ℎ,𝐵^ℎ−Φ^ℎ)∈𝑍_ℎ×𝑋_2ℎ and (5.9), we get 𝑎_0ℎ+𝑎𝑤^ℎ_1ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ+𝑎_1ℎ𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ,(𝑢,𝐵),𝑣^ℎ,Ψ^ℎ=𝑎_0ℎ𝑢^ℎ,𝐵^ℎ,𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ+𝑏_ℎ𝑣^ℎ,Ψ^ℎ,𝑠^ℎ−𝑝^ℎ+𝑎_1ℎ𝑤^ℎ,Φ^ℎ−(𝑢,𝐵),𝑣^ℎ,Ψ^ℎ+𝑎_1ℎ𝑤^ℎ,Φ^ℎ−(𝑢,𝐵),(𝑢,𝐵),𝑣^ℎ,Ψ^ℎ+𝐸𝑣_ℎ𝑢^ℎ,𝐵^ℎ,𝑤^ℎ,Φ^ℎ−(𝑢,𝐵),𝑣^ℎ,Ψ^ℎ+𝑏_ℎ𝑣^ℎ,Ψ^ℎ,𝑠^ℎ−𝑝,Ψ^ℎ.(5.12) Notice that 𝑎_1ℎ𝑢^ℎ,𝐵^ℎ𝑏,𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ,𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ=0,_ℎ𝑢^ℎ−𝑤^ℎ,𝐵^ℎ−Φ^ℎ,𝑠^ℎ−𝑝^ℎ=0.(5.13) Let (𝑣^ℎ,Ψ^ℎ)=(𝑤^ℎ,Φ^ℎ)−(𝑢^ℎ,𝐵^ℎ), and by (5.12), we obtain 𝑎_0ℎ+𝑎𝑤^ℎ_1ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ,𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ=𝑎_0ℎ𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ,(𝑢,𝐵),𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ+𝑎_1ℎ𝑤^ℎ,Φ^ℎ−(𝑢,𝐵),𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ+𝑎_1ℎ𝑤^ℎ,Φ^ℎ−(𝑢,𝐵),(𝑢,𝐵),𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ+𝑏_ℎ𝑢^ℎ,𝐵^ℎ,𝑤^ℎ,Φ^ℎ−(𝑢,𝐵),𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ,𝑠^ℎ−𝑝+𝐸𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ.(5.14) Using the continuity properties of 𝑎_0ℎ,𝑎_1ℎ and the stability bounds for ‖(𝑢,𝐵)‖_𝑊 and ‖(𝑢^ℎ,𝐵^ℎ)‖_ℎ in Theorems 2.1 and 4.6, respectively, the right-hand side of (5.14) can be bounded by r.h.s.≤‖‖(𝑤^ℎ,Φ^ℎ)−(𝑢^ℎ,𝐵^ℎ)‖‖_ℎ×𝐶𝛾₂‖‖(𝑤^ℎ,Φ^ℎ)−(𝑢,𝐵)‖‖_ℎ+𝐶_𝑐‖‖(𝑤^ℎ,Φ^ℎ)−(𝑢,𝐵)‖‖_ℎ‖(𝑢,𝐵)‖_𝑊+𝐶_𝑐‖‖(𝑤^ℎ,Φ^ℎ)−(𝑢,𝐵)‖‖_ℎ‖‖(𝑢^ℎ,𝐵^ℎ)‖‖_ℎ+𝐶‖‖𝑠^ℎ−𝑝‖‖₀+𝐸𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ≤𝐶‖‖(𝑤_ℎ‖‖(𝑤^ℎ,Φ^ℎ)−(𝑢^ℎ,𝐵^ℎ)‖‖_ℎ,Φ^ℎ)−(𝑢^ℎ,𝐵^ℎ)‖‖_ℎ‖‖(𝑤^ℎ,Φ^ℎ)−(𝑢,𝐵)‖‖_ℎ+‖‖𝑠^ℎ−𝑝‖‖₀+𝐸𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ‖‖𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ‖‖_ℎ.(5.15) Next, the coercivity property of the form 𝑎_0ℎ, continuity of 𝑎_1ℎ in Lemma 4.4, stability bound for ‖(𝑢,𝐵)‖_𝑊 in Theorem 2.1, and the assumption 𝐶_𝑐𝛾₃‖𝑓‖₋₁/𝐶_𝑎𝐶₁𝛾²¹<1/2 allow us to bound the left-hand side of (5.14) as l.h.s.≥𝐶_𝑎𝛾₁‖‖(𝑤^ℎ,Φ^ℎ)−(𝑢^ℎ,𝐵^ℎ)‖‖^2ℎ−𝐶_𝑐𝛾₃‖‖(𝑤^ℎ,Φ^ℎ)−(𝑢^ℎ,𝐵^ℎ)‖‖^2ℎ‖(𝑢,𝐵)‖_𝑊≥12𝐶^𝑎𝛾₁‖‖(𝑤^ℎ,Φ^ℎ)−(𝑢^ℎ,𝐵^ℎ)‖‖^2ℎ.(5.16) Combining these bounds, we have ‖‖(𝑤^ℎ,Φ^ℎ)−(𝑢^ℎ,𝐵^ℎ)‖‖_ℎ≤𝐶‖‖𝑤^ℎ,Φ^ℎ−(𝑢,𝐵)‖‖_ℎ+‖‖𝑠^ℎ−𝑝‖‖₀+𝐸𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ‖‖𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ‖‖_ℎ.(5.17) Then, applying the triangle inequality, we get ‖‖(𝑢,𝐵)−(𝑢^ℎ,𝐵^ℎ)‖‖_ℎ≤𝐶‖‖𝑤^ℎ,Φ^ℎ−(𝑢,𝐵)‖‖_ℎ+‖‖𝑠^ℎ−𝑝‖‖₀+𝐸𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ‖‖𝑤^ℎ,Φ^ℎ−𝑢^ℎ,𝐵^ℎ‖‖_ℎ.(5.18) Now, for (𝑤^ℎ,Φ^ℎ)∈𝑍_ℎ×𝑋_2ℎ,𝑠^ℎ∈𝑀_ℎ, taking the infimum of (5.18) yields ‖‖(𝑢,𝐵)−(𝑢^ℎ,𝐵^ℎ)‖‖_ℎ≤𝐶⎡⎢⎢⎣inf_{𝑤ℎ,Φℎ∈𝑍ℎ×𝑋2ℎ}‖‖𝑤^ℎ,Φ^ℎ−(𝑢,𝐵)‖‖_ℎ+inf_{𝑠ℎ∈𝑀ℎ}‖‖𝑠^ℎ−𝑝‖‖₀+sup_{𝑣ℎ,Ψℎ∈𝑍ℎ×𝑋2ℎ}𝐸𝑣^ℎ,Ψ^ℎ‖‖𝑣^ℎ,Ψ^ℎ‖‖_ℎ⎤⎥⎥⎦.(5.19) With the argument as [37], we know that inf_{(𝑤ℎ,Φℎ)∈𝑍ℎ×𝑋2ℎ}‖‖(𝑤^ℎ,Φ^ℎ)−(𝑢,𝐵)‖‖_ℎ≤𝐶inf_{(𝑣ℎ,Ψℎ)∈𝑊ℎ}‖‖(𝑤^ℎ,Φ^ℎ)−(𝑢,𝐵)‖‖_ℎ.(5.20) Substituting (5.20) into (5.19) implies (5.2).Step 2. For (𝑣^ℎ,Ψ^ℎ)∈𝑊_ℎ,𝑠^ℎ∈𝑀_ℎ, we have from (5.9) that 𝑏_ℎ𝑣^ℎ,Ψ^ℎ,𝑠^ℎ−𝑝^ℎ==𝑏𝑏_ℎℎ𝑣^ℎ,Ψ^ℎ,𝑠^ℎ−𝑝+𝑏_ℎ𝑣^ℎ,Ψ^ℎ,𝑝−𝑝^ℎ−𝑎_1ℎ𝑣^ℎ,Ψ^ℎ,𝑠^ℎ−𝑝−𝑎_0ℎ(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ−𝑎_1ℎ(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,(𝑢,𝐵),𝑣^ℎ,Ψ^ℎ+𝐸𝑣_ℎ𝑢^ℎ,𝐵^ℎ,(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ,Ψ^ℎ.(5.21) Using the continuity properties of 𝑎_0ℎ and 𝑎_1ℎ and the discrete inf-sup condition (4.21) of Lemma 4.5, it follows that ‖‖𝑠^ℎ−𝑝^ℎ‖‖₀≤1𝛽^⋆𝐶‖‖𝑠^ℎ−𝑝‖‖₀+𝐶𝛾₂+𝐶_𝑐‖(𝑢,𝐵)‖_𝑊+‖‖𝑢^ℎ,𝐵^ℎ‖‖_ℎ‖‖(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ‖‖_ℎ+𝐸𝑣^ℎ,Ψ^ℎ‖‖𝑣^ℎ,Ψ^ℎ‖‖_ℎ(5..22) Then, with the help of the triangle inequality and (5.2), we complete the proof.

Theorem 5.2. Let 𝑢∈(𝐻¹⁰(Ω)³∩𝐻²(Ω)³), 𝐵∈(𝐻²(Ω)³∩𝐻^1𝑛(Ω)³), 𝑝∈(𝐿²⁰(Ω)∩𝐻¹(Ω)), and ((𝑢^ℎ,𝐵^ℎ),𝑝^ℎ)∈𝑊_ℎ×𝑀_ℎ be the solutions of Problems (𝐈₁) and (𝐈₂), respectively. Then there holds ‖‖(𝑢,𝐵)−(𝑢^ℎ,𝐵^ℎ)‖‖_ℎ+‖‖𝑝−𝑝^ℎ‖‖₀≤𝐶ℎ|𝑢|₂+‖𝐵‖₂+‖𝑝‖₁.(5.23)

Proof. On the one hand, the interpolation theory gives inf_{𝑣ℎ∈𝑋1ℎ}‖‖𝑢in−f𝑣^ℎ_{Ψℎ∈𝑋2ℎ}‖‖^21ℎ≤‖‖𝑢−Π¹𝑢‖‖^21ℎ≤𝐶ℎ²|𝑢|²²,‖‖𝐵−Ψ^ℎ‖‖²¹≤𝐶ℎ²‖𝐵‖²².(5.24) Therefore, by (5.24), we obtain inf_{(𝑣ℎ,Ψℎ)∈𝑊ℎ}‖‖(𝑢,𝐵)−(𝑣^ℎ,Ψ^ℎ)‖‖_ℎ≤𝐶ℎ|𝑢|₂+‖𝐵‖₂.(5.25) At the same time, for 𝑝∈𝐿²⁰(Ω), we define the interpolation 𝑅^ℎ0𝑝∈𝑀_ℎ on each element 𝐾 as _𝐾𝑝−𝑅^ℎ0𝑝𝑑𝐱=0.(5.26) Then there holds inf_{𝑠ℎ∈𝑀ℎ}‖‖𝑝−𝑠^ℎ‖‖₀≤‖‖𝑝−𝑅^ℎ0𝑝‖‖₀≤𝐶ℎ‖𝑝‖₁.(5.27) On the other hand, by the similar techniques to [25–27, 29, 30, 32], we have ||𝐸𝑣^ℎ,Ψ^ℎ||≤𝐶ℎ|𝑢|₂+‖𝑝‖₁‖‖(𝑣^ℎ,Ψ^ℎ)‖‖_ℎ.(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. 𝑅−𝑀⁻²Δ𝑤+𝑁⁻¹[𝑤⋅∇𝑢−𝑢⋅∇𝑤]+∇𝑠+𝑅^𝑚−1(∇×Φ)×𝐵=𝑢−𝑢^ℎ,inΩ,_𝑚−2[=∇𝐵×−(∇𝐵×_ℎΦ)−∇(∇⋅Φ)]+𝑅^𝑚−1[(∇×𝐵)×𝑤−(∇×Φ)×𝑢−∇×(𝐵×𝑤)]∇𝑤𝐵⋅⋅=𝑛𝑤0==,00,o,ni𝜕n𝑅Ω,Ω,_𝑚−1,inΩ,(∇×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 𝑠∈𝐿²⁰(Ω) such that for all (𝑣,Ψ)∈𝑊(Ω),𝜓∈𝐿²⁰(Ω)𝑎₀((𝑣=,Ψ()𝑢,,(𝐵𝑤),−Φ))𝑢+_ℎ𝑎₁((𝑢,𝐵),(𝑣,Ψ),(𝑤,Φ))+𝑎₁((𝑣,Ψ),(𝑢,𝐵),(𝑤,Φ))+𝑏((𝑣,Ψ),𝑠)𝑏((𝑤,Φ),𝜓)=0.,𝐵^ℎ,(𝑣,Ψ),(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+‖𝑠‖₁≤𝐶‖‖(𝑢,𝐵)−(𝑢^ℎ,𝐵^ℎ)‖‖₀.(5.31) Let ((𝑤^ℎ,Φ^ℎ),𝑠^ℎ)∈𝑊_ℎ×𝑀_ℎ satisfy ‖‖(𝑤,Φ)−(𝑤^ℎ,Φ^ℎ)‖‖_ℎ+‖‖𝑠−𝑠^ℎ‖‖₀≤𝐶ℎ|𝑤|₂+‖Φ‖₂+‖𝑠‖₁.(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^ℎ,𝐵^ℎ)‖‖₀≤𝐶ℎ²|𝑢|₂+‖𝐵‖₂+‖𝑝‖₁.(5.33)

Proof. By (5.31) and (5.32), we deduce that ‖‖(𝑤,Φ)−𝑤^ℎ,Φ^ℎ‖‖_ℎ+‖‖𝑠−𝑠^ℎ‖‖₀≤𝐶ℎ‖‖(𝑢,𝐵)−(𝑢^ℎ,𝐵^ℎ)‖‖₀.(5.34) Multiplying (𝑢−𝑢^ℎ) and (𝐵−𝐵^ℎ) 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^ℎ,𝐵^ℎ)‖‖²⁰=+𝑎𝑎_0ℎ_1ℎ(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,(𝑤,Φ)+𝑎_1ℎ(𝑢,𝐵),(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,(𝑤,Φ)−(2𝑁)(𝑢₋₁,𝐵)−𝑢^ℎ,𝐵^ℎ,(𝑢,𝐵),(𝑤,Φ)+𝑏_ℎ(𝑢,𝐵)−u^ℎ,𝐵^ℎ,𝑠_𝐾_∈Γℎ_𝐾div𝑢−𝑢^ℎ(𝑢⋅𝑤)𝑑𝐱+𝐹𝑢−𝑢^ℎ,(5.35) where 𝐹𝑢−𝑢^ℎ=−𝑀⁻²_𝐾_∈Γℎ_𝜕𝐾𝜕𝑤𝜕𝑛𝑢−𝑢^ℎ𝑑𝑠−(2𝑁)⁻¹_𝐾∈Γℎ_𝜕𝐾+(2𝑁)₋₁(𝑢⋅𝑛)𝑤⋅𝑢−𝑢^ℎ𝑑𝑠_𝐾_∈Γℎ_𝜕𝐾𝑢−𝑢^ℎ⋅𝑛(𝑢⋅𝑤)𝑑𝑠+_𝐾∈Γℎ_𝜕𝐾𝑠𝑢−𝑢^ℎ⋅𝑛𝑑𝑠.(5.36) Subtraction of (5.9) yields 𝑎_0ℎ+𝑎(𝑢,_1ℎ𝐵)−𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ+𝑎_1ℎ(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,(𝑢,𝐵),𝑣^ℎ,Ψ^ℎ=𝐸𝑢𝑣^ℎ_ℎ,𝐵^ℎ,(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,𝑣^ℎ,Ψ^ℎ+𝑏_ℎ𝑣^ℎ,Ψ^ℎ,𝑝−𝑝^ℎ,Ψ^ℎ.(5.37) Note that 𝑏_ℎ(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,𝜙^ℎ=0,∀𝜙^ℎ∈𝑀_ℎ.(5.38) Now, setting 𝜓=𝑝−𝑝^ℎ in Problem (𝐈_𝟑), we have 𝑏(𝑤,Φ),𝑝−𝑝^ℎ=0.(5.39) From (5.35)–(5.39), we get ‖‖(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ‖‖²⁰=+𝑎𝑏_0ℎℎ(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,(𝑤,Φ)−𝑣^ℎ,Ψ^ℎ+𝑏_ℎ(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,𝑠−𝜙^ℎ(𝑤,Φ)−𝑣^ℎ,Ψ^ℎ,𝑝−𝑝^ℎ+𝐴₁+𝐴₂+𝐴₃,(5.40) where 𝐴₁=+𝑎𝑎_1ℎ_1ℎ𝑢^ℎ,𝐵^ℎ,(𝑢,𝐵)−𝑢^ℎ,B^ℎ,(𝑤,Φ)−𝑣^ℎ,Ψ^ℎ+𝑎_1ℎ(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,(𝑢,𝐵),(𝑤,Φ)−𝑣^ℎ,Ψ^ℎ𝐴(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,(𝑤,Φ)₂,𝐴=𝐹𝑢−𝑢^ℎ+𝐸𝑣^ℎ,Ψ^ℎ₃,=−12𝑁_𝐾∈Γℎ_𝐾div𝑢−𝑢^ℎ(𝑢⋅𝑤)𝑑𝐱(5.41) By (2.8), Lemma 4.4, and Theorem 4.6, we find 𝐴₁≤𝐶‖‖(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ‖‖_ℎ‖‖(𝑤,Φ)−𝑣^ℎ,Ψ^ℎ‖‖_ℎ+‖‖(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ‖‖^2ℎ‖(𝑤,Φ)‖_𝑊.(5.42) From [45], we know 𝐹𝑢−𝑢^ℎ≤𝐶ℎ|𝑤|₂+‖𝑠‖₁‖‖𝑢−𝑢^ℎ‖‖_1ℎ.(5.43) By virtue of 𝑢,𝑤∈𝐻²(Ω)³↪𝐶⁰(Ω)², we obtain 𝐸=𝑣^ℎ,Ψ^ℎ_𝐾∈Γℎ_𝜕𝐾𝑀⁻²𝜕𝑢𝜕𝑛𝑣^ℎ≤𝐶ℎ|𝑢|₂−𝑤−𝑝𝑣^ℎ−𝑤⋅𝑛−(2𝑁)⁻¹(𝑢⋅𝑛)𝑢⋅𝑣^ℎ−𝑤𝑑𝑠+‖𝑝‖₁‖‖𝑣^ℎ−𝑤‖‖_1ℎ.(5.44) Let 𝑎_𝐾 be a constant such that ‖‖𝑢⋅𝑤−𝑎_𝐾‖‖_0,𝐾≤𝐶ℎ‖𝑢⋅𝑤‖_1,𝐾≤𝐶ℎ‖𝑢‖_1,𝐾‖𝑤‖_2,𝐾.(5.45) Since div𝑢=0,𝑏_ℎ((𝑢^ℎ,𝐵^ℎ),𝑞)=0, ∀𝑞∈𝑀_ℎ and (5.45), we obtain ||𝐴₃||=|||||−12𝑁_𝐾∈Γℎ_𝐾≤𝐶ℎ‖‖𝑢−𝑢_ℎdiv𝑢−𝑢^ℎ𝑢⋅𝑤−𝑎_𝐾𝑑𝐱|||||‖‖_1ℎ‖𝑢‖₁‖𝑤‖₂.(5.46) Thus, by (5.31) and the approximation theory, there hold inf_{(𝑣ℎ,Ψℎ)∈𝑊ℎ}‖‖i(n𝑤f,_{𝑣ℎ∈𝑋2ℎ}Φ)−(𝑣^ℎ,Ψ^ℎ)‖‖_ℎ≤𝐶ℎ‖(𝑤,Φ)‖₂≤𝐶ℎ‖‖(𝑢,𝐵)−(𝑢^ℎ,𝐵^ℎ)‖‖₀,inf‖‖_{𝜙ℎ∈𝑀ℎ}𝑤−𝑣^ℎ‖‖_1ℎ≤𝐶ℎ‖𝑤‖₂≤𝐶ℎ‖‖(𝑢,𝐵)−(𝑢^ℎ,𝐵^ℎ)‖‖₀,‖(𝑤‖‖,Φ𝑠−)‖𝜙^ℎ_𝑊‖‖₀≤𝐶ℎ‖𝑠‖₁≤𝐶ℎ‖‖(𝑢,𝐵)−(𝑢^ℎ,𝐵^ℎ)‖‖₀,||𝑏_ℎ≤‖(𝑤,Φ)‖₂≤𝐶‖‖(𝑢,𝐵)−(𝑢^ℎ,𝐵^ℎ)‖‖₀,||𝑏_ℎ(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ,𝑠−𝜙^ℎ||≤𝐶‖‖(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ‖‖_ℎ‖‖𝑠−𝜙^ℎ‖‖₀,(𝑤,Φ)−𝑣^ℎ,Ψ^ℎ,𝑝−𝑝^ℎ||≤𝐶‖‖(𝑢,𝐵)−𝑢^ℎ,𝐵^ℎ‖‖_ℎ‖‖𝑠−𝜙^ℎ‖‖₀.(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 𝑢∈𝐻¹⁰(Ω)³,𝐵∈𝐻={𝐵∈𝐻¹(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).