Fully Discrete Finite Element Approximation of the MHD Flow

1 Introduction

This work is concerned with the following 3D incompressible magnetohydrodynamic (MHD) equations under the influence of body forces [11, 21, 28, 3, 25]:

where Ω ⊂ R 3 is an open bounded domain with a smooth boundary, 𝐮, 𝐁 and 𝑝 denote the velocity field, the magnetic field and the pressure, respectively. The three parameters 𝜈, 𝜇 and 𝑠 are given by ν = Re - 1 , μ = Re m - 1 , s = M 2 / ( Re ⁢ Re m ) , respectively, with Re and Re m being the Reynolds number and the magnetic Reynolds number, and M > 0 being the Hartman number. For convenience, we shall often write r = ( x , y , z ) ∈ Ω , and the magnetic field 𝐁 as B ⁢ ( t ) or B ⁢ ( r , t ) ; the same for the pressure 𝑝 and velocity 𝐮.

System (1.1) couples the incompressible Navier–Stokes equations with Maxwell’s equations, and is considered in conjunction with the following initial boundary conditions [11, 21, 28, 3, 25]:

where ∇ ⋅ u 0 ⁢ ( r ) = 0 and ∇ ⋅ B 0 ⁢ ( r ) = 0 , with 𝐧 being the unit outward normal vector of ∂ ⁡ Ω .

We will frequently use the following spaces in our subsequent analysis:

as well as the following trilinear forms:

For the coupled MHD flow system (1.1)–(1.2), we are interested in its variational formulation: find ( u ⁢ ( t ) , p ⁢ ( t ) , B ⁢ ( t ) ) ∈ X × M × W such that it satisfies for all ( v , q , C ) ∈ X × M × W the system

There are wide studies of the stability and convergence of the fully discrete second-order schemes based on the finite element spatial discretization for solving the time-dependent Navier–Stokes equations. For further exposition, we let 0 < h < 1 be the spatial mesh size and 0 < τ = T N < 1 a time step size, and we let u h m , B h m and p h m be the finite element approximate solutions of 𝐮, 𝐁 and 𝑝 at t = t m , respectively. We know that, for second-order or higher-order schemes for solving the time-dependent Navier–Stokes equations when the nonlinear term is treated explicitly, the stability or convergence is often achieved under the conditions of the form τ ⁢ h - α ≤ C 0 for some α > 0 ; see, for example, [23, 2, 20, 29]. These conditions can be improved, e.g., in [13], where the Adams–Bashforth and Crank–Nicolson schemes were considered in combination with a finite element spatial discretization for solving the 2D time-dependent Navier–Stokes equations. It was proved in [13] that the stability and convergence are guaranteed under the condition τ ≤ C 0 and with the following error estimates:

for all t m ∈ ( 0 , T ] . Here σ ⁢ ( t ) = min ⁡ { 1 , t } , and 𝜅 is a generic positive constant depending on the data 𝜈, Ω, 𝑇, u 0 and 𝑓. Hereafter, we use 𝜅 to denote a generic positive constant depending on the data 𝜈, 𝜇, 𝑠, Ω, 𝑇, u 0 , B 0 and 𝑓. It was further shown in [15] that the convergence and stability of the Euler semi-implicit scheme for the 3D MHD equations can be also ensured under the condition τ ≤ C 0 .

In this work, we consider the Adams–Bashforth and Crank–Nicolson schemes for the time discretizations of the nonlinear and linear terms, respectively, and the finite element method for the spatial discretization for solving the time-dependent coupled MHD flow system (1.1)–(1.2) in three dimensions. Due to the explicit treatment of the nonlinear terms in the model, we cannot achieve the desired stability estimates of the fully discrete finite element solutions; therefore, it is hard for us to derive the optimal error estimates by means of existing approaches for both the Navier–Stokes equations and the coupled MHD flow system. We shall overcome this technical difficulty by a delicate induction method. Moreover, we are able to establish the optimal second-order convergence of the numerical velocity and magnetic field in the L 2 -norm by using a special negative-norm technique, while the standard discrete duality strategy [19, 30, 12, 14, 13] does not work.

The rest of the work is organized as follows. In § 2, some basic assumptions and inequalities related to the MHD flow and the basic properties of the finite element solution for the 3D MHD flow are presented. In § 3, we provide some estimates and smoothing properties of the finite element solution ( u h , p h , B h ) . In § 4, we develop a fully discrete finite element approximation based on the Crank–Nicolson/Adams–Bashforth scheme for the 3D MHD equations and then derive its truncation error estimates in § 5. In § 6, we establish the optimal error estimates for the numerical solution ( u h n , B h n , p h n ) and present some numerical experiments in § 7 to verify the convergence orders of the numerical scheme.

2 Finite Element Spatial Discretization for the MHD Flow

In this section, we discuss the finite element spatial discretization for the MHD equations (1.1)–(1.2). To do so, we first introduce a triangulation of the domain Ω. For the sake of technical treatments, we assume that the boundary of domain Ω is a closed convex polyhedron; the actual curved boundary case can be treated using some well-developed technicalities for the curved boundary in combination with the finite element error estimates established in the current work. Let T h be a triangulation of the polyhedral domain Ω, and let X h ⊂ X , M h ⊂ M = L 0 2 ⁢ ( Ω ) and W h ⊂ W be some continuous piecewise finite element spaces defined on T h . We shall also need the following subspace of X h :

The finite element spaces X h ⊂ X , M h ⊂ M = L 0 2 ⁢ ( Ω ) and W h ⊂ W will be made clearer later.

Throughout this paper, we make the following two assumptions on the prescribed data and the solutions to system (1.4)–(1.5), which specify the regularities of the data and solutions needed for our main results. Hereafter, 𝑐, 𝜅 and c i , κ i for i ≥ 0 are some generic positive constants depending only on some of the given data 𝜈, 𝜇, 𝑠, Ω, 𝑇, u 0 , B 0 and 𝐟.

Also, we assume that the domain Ω is smooth to ensure the following estimates [4, 17, 7, 6, 26].

We shall often use the Stokes operator A 1 = - P ⁢ Δ and the Maxwell operator A 2 = P ∇ × ∇ × , with

Here 𝑃 is an L 2 -projection from L 2 ⁢ ( Ω ) 3 into 𝐇.

Now we make a standard approximation assumption on the finite element space system ( X h , M h , W h ) [1, 5, 10, 17, 18, 24, 9, 27].

Here is an example of the finite element spaces ( X h , M h , W h ) that satisfy Assumption (A3) above (cf. [3, 25]):

Now we formulate the semi-discrete finite element approximation of system (1.4)–(1.5), whose properties will be crucial to our error estimates of the finite element solution to the fully discrete scheme: find ( u h ⁢ ( t ) , p h ⁢ ( t ) , B h ⁢ ( t ) ) ∈ X h × M h × W h such that it satisfies, for all ( v h , q h , C h ) ∈ X h × M h × W h ,

and the A h -induced discrete norm ∥ v h ∥ α = ∥ A h α 2 ⁢ v h ∥ 0 , Ω for α ∈ R . In particular, we have

Similarly to the discrete Stokes operator A h , we define the discrete Maxwell operator A 2 ⁢ h as follows: for any B h ∈ W h , A 2 ⁢ h ⁢ B h = R 0 ⁢ h ⁢ ( ∇ h × ∇ × B h + ∇ h ⁡ ∇ ⋅ B h ) ∈ W h , satisfying

We will also use the A 2 ⁢ h -induced discrete norm ∥ B h ∥ α = ∥ A 2 ⁢ h α 2 ⁢ B h ∥ 0 , Ω for any α ∈ R . Then we have

We end this section by recalling some stability and error estimates of the solution ( u ⁢ ( t ) , p ⁢ ( t ) , B ⁢ ( t ) ) to system (1.4)–(1.5) and the numerical solution ( u h ⁢ ( t ) , p h ⁢ ( t ) , B h ⁢ ( t ) ) to the semi-discrete system (2.1)–(2.2) [16].

3 Basic Estimates and Smoothing Properties of Discrete Solutions

In this section, we present some discrete inequalities of finite element functions and some smoothing properties of the finite element solution ( u h , p h , B h ) to the semi-discrete system (2.1)–(2.2). First, we recall some discrete Gadliardo–Nirenberg estimates [17, 18]:

By using Assumptions (A2)–(A3) and some similar arguments to the ones used by Heywood and Rannacher in [17], we can prove the following discrete Gadliardo–Nirenberg estimates.