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

The nonconformingmixed 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 H1norm 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.


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 H 1 -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 P 1 /P 0 element and the rotated Q 1 /Q 0 element and got first-order accuracy, respectively.Reference 27 modified the rotated Q 1 /Q 0 element used in 26 and derived the same convergence order as 26 .For the Navier-Stokes equations, 28-30 obtained maximum norm estimates of P 1 /P 0 element and the optimal error estimates of EQ rot 1 /Q 0 element both in broken H 1 -norm for the velocity field and in L 2 -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 H 1 -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 L 2 -norm are established, as well as those in a broken H 1 -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 L 2 -norm are established, as well as ones in broken H 1 -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.

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 u 1 , u 2 , u 3 , the pressure p, the magnetic field where Ω is a simply connected, bounded domain with unit outward normal n n 1 , n 2 , n 3 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.
here and later, x x, y, z .
The mixed variational formulation for Problem I is written as follows. where

2.4
It has been shown in 9, 37, 38 that for u, v, w respectively, where The following result can be found in 9 .
Theorem 2.1.If f ∈ H −1 Ω 3 , then Problem I 1 has at least a solution, in addition, that is unique provided that and satisfying the stability bound where are positive constants only depending on the domain Ω.

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 n Ω 3 to approximate the unknown variables u, p, and B. The following assumptions about the space pair X 1h , M h are provided: , 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, Π 1 is the interpolation operator associated with X 1h satisfying Π K Π 1 | K for K ∈ Γ h , and P 1 K is the polynomial space of degree less than or equal to one on K.
Introduce the finite element space

3.1
The finite element space X 2h is defined by where Q 1 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.

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 Then the approximate formulation of Problem I 1 reads as follows.

4.8
From the definition of 4.3 , a 1h satisfies the following antisymmetric properties 9 : 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: Proof.We consider the following problem:

4.12
Then by 3 , the solution B of 4.12 satisfies On the one hand, by Green's formula and H ölder's inequality, we deduce that

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 Proof.The first result is wellknown 30, 37, 38 .To prove the second result, we need the imbedding properties H 1 n Ω 3 → H 1 Ω 3 → L 4 Ω 3 and the discrete imbedding inequality showed in 30, 32 : 4.15 Thus, 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: where C c , C a are positive constants, independent of h.
Proof.Firstly, using the triangle inequality and Lemma 4.3 yields

4.19
With the help of H ölder's inequality, we find

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 Proof.On the one hand, by 37, 38 , there exists a constant β > 0 such that inf Therefore, by the assumption E and 4.22 , we obtain sup where β β/C > 0. The proof is completed.
From Lemmas 4.4-4.5, we have the following.

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 H 1 -norm.

5.1
Let u, B , p ∈ W Ω × L 2 0 Ω and u h , B h , p h ∈ W h × M h be the solutions of Problems I 1 and I 2 , respectively.Then there hold

5.3
where Proof.We proceed in two steps.
Step 1.For v h , Ψ h ∈ W h , by Green's formula, we have

5.6
Here, we have used the following equality: On the other hand, we have from 4.8

5.8
Subtraction of 4.8 from 5.6 yields

5.9
Let w h , Φ h be an arbitrary element of Z h × X 2h , that is: Then, and 5.9 , we get

5.12
Notice that

5.13
Journal of Applied Mathematics 13 Let v h , Ψ h w h , Φ h − u h , B h , and by 5.12 , we obtain

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

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 c γ 3 f −1 /C a C 1 γ 2 1 < 1/2 allow us to bound the left-hand side of 5.14 as

5.16
Combining these bounds, we have

5.17
Then, applying the triangle inequality, we get

5.18
Now, for w h , Φ h ∈ Z h × X 2h , s h ∈ M h , taking the infimum of 5.18 yields

5.19
With the argument as 37 , we know that inf
Step 2. For v h , Ψ h ∈ W h , s h ∈ M h , we have from 5.9 that

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

5.22
Then, with the help of the triangle inequality and 5.2 , we complete the proof.
, and u h , B h , p h ∈ W h × M h be the solutions of Problems I 1 and I 2 , respectively.Then there holds

5.23
Proof.On the one hand, the interpolation theory gives

5.24
Therefore, by 5.24 , we obtain inf

5.25
At the same time, for p ∈ L 2 0 Ω , we define the interpolation R h 0 p ∈ M h on each element K as Then there holds inf

5.27
On the other hand, by the similar techniques to 25-27, 29, 30, 32 , we have

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 L 2 -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.

5.29
The variational formulation of 5.29 is written as follows.

5.30
Under the same hypotheses as Theorem 2.1, we may easily know that Problem I 3 has a unique solution w, Φ , s ∈ W Ω × L 2 0 Ω .We require that 5.29 be H 2 -regular, that is:

5.42
From 45 , we know

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 div u 0, b h u h , B h , q 0, ∀q ∈ M h and 5.45 , we obtain

5.46
Thus, by 5.31 and the approximation theory, there hold inf

5.47
Combining these inequalities and using Lemma 4.4 and the results from 5.39 to 5.46 yields the desired result.

1h≤ Ch w 2 ≤≤ Ch s 1 ≤
Ch u, B − u h , B h Ch u, B − u h , B h 0 , w, Φ W ≤ 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 , b h w, Φ − v h , Ψ h , p − p h ≤ C u, B − u h , B h h s − φ h 0 .
− 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 5.3.Under the hypothesis of Theorem 5.2, let w, Φ , s be the solution of Problem I 3 , and assume that 5.31 holds.Then we have − u h , B h , v h , Ψ h a 1h u, B − u h , B h , u, B , v h , Ψ h a 1h u h , B h , u, B − u h , B h , v h , Ψ h b h v h , Ψ h , p − p h E v h , Ψ h .−uh, B h , w, Φ − v h , Ψ h b h u, B − u h , B h , s − φ h b h w, Φ − v h , Ψ h , p − p h A 1 A 2 A 3 ,5.40whereA 1 a 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 1h u, B − u h , B h , u, B − u h , B h , w, Φ , 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