A Linear Stabilized Incompressible Magnetohydrodynamic Problem with Magnetic Pressure

Abstract

The objective of this article is to examine, stabilize, and linearize the incompressible magnetohydrodynamic model equations. The approximate solutions are carried out through the lowest equal order mixed finite element (FE) approach, involving variables such as fluid velocity, hydro pressure, magnetic field, and magnetic pressure. The formulation of the variational form for the approximate solution necessitates the use of a pair of approximating spaces. However, these spaces cannot be arbitrarily chosen; they must adhere to strict stability conditions, notably the Ladyzhenskaya–Babuska–Brezzi (LBB) or inf-sup condition. This study addresses the absence of stabilization and linearization techniques in the incompressible magnetohydrodynamic model equations using the lowest equal order mixed finite element approach. The article introduces a stabilization technique to meet two stability conditions, proving its existence and uniqueness. This novel approach was not previously explored in the literature. The proposed stabilized technique does not necessitate parameters or computing higher-order derivatives, making it computationally efficient. The study offers numerical tests demonstrating optimal convergence and effectiveness of the revised approach in two-dimensional settings.

1. Introduction

Magnetohydrodynamical (MHD) model equations are commonly used to investigate the dynamic properties of conducting fluids, including electrolytes, liquid metals, and plasmas. Because applied magnetic fields may create currents, which polarize the fluid and change the magnetic field in return, this system couples with hydrodynamics (Navier–Stokes equations) and electromagnetism (Maxwell’s equations) [1,2,3,4].

These equations are used in many industrial processes, such as continuous steel casting, MHD generators, nuclear fusion reactor crustal growth devices, and MHD pumps that use conduction or induction principles. Moreover, the shortage of experimental data and the complicated nature of the design have created an increasing need for numerically stabilized approaches for this equation system. Many studies have investigated various stabilization techniques, including convection-type stabilization methods [5], DG methods [6], least square methods, lowest equal order methods, weighted exact penalty formulation methods [7], exact penalty formulation methods [8], two-level stabilization methods [1,9], least square method [10], and others. Though significant progress has been made in developing approaches for complex algebraic problems, solving them numerically is still a difficult and costly process.

There are several recognized difficulties with the Galerkin finite element approach to this issue. Initially, oscillations might occur, especially when first-order derivatives dominate second-order effects in the Navier–Stokes equations. Secondly, even though the continuous version of the scheme satisfies compatibility conditions, this does not ensure that the discrete version will meet the inf-sup standards. This result depends on which finite element spaces, $\mathcal{X}$, $\mathcal{Q}$, $\mathcal{W}$, and $\mathcal{M}$, are chosen. After all, a discrete system of equations solution may result in numerical complexity when the two problems interact strongly, especially when it comes to combining the hydrodynamical and electromagnetic elements.

However, because they are easy to build and provide enough precision for a wide range of applications, low-order finite elements, including $p_0$ and $p_1$ elements, are frequently chosen by practitioners as a solution to these problems. Furthermore, because of their lack of stability, the finite element pairings $p_1-p_0$ and $p_1-p_1$ are unacceptable for the Stokes equations. The FE spaces between $\mathcal{X}$ and $\mathcal{Q}$, or $\mathcal{W}$ and $\mathcal{M}$, do not satisfy the well-known inf-sup condition for these simple and linear elements. To implement the stabilized scheme, projection operators are used, which only need conventional nodal data structures and can be easily evaluated at the element level. In order to solve the above-described problems, this work aims to develop a stabilization mechanism. We refer to [11,12] for the related stabilization technique, which has been implemented in the literature for different model issues.

As mentioned in [13], the stabilization method used in this investigation was initially developed to handle a single inf-sup condition in the context of the Stokes model. The difficulties arising from the strong interaction between hydrodynamical and electromagnetic issues, which caused numerous numerical issues in addressing the discrete system, made this change imperative. Notably, proving the compliance of the two critical qualities (two inf-sup criteria) required for the mixed finite element approach is a major problem in assuring the well-posedness of the novel scheme. To ensure that the linked model equations are well-posed, these requirements are necessary. The key to overcoming such a challenge is to include a pressure stabilization term in the variational formulation. The implementation of stabilization techniques in combination with coupling equations in the viscoelastic field has been the subject of recent finite element methods research (see [11,12]).

This study aims to utilize the lowest equal order finite elements collectively with two stabilization terms to approximate the solution, which will have the advantages of being easy to write into existing computer code, not requiring the solution of second-order derivatives or above, and being effective without parameters. The innovation of this paper lies in its development of a novel stabilization technique for the complex nonlinear incompressible magnetohydrodynamic (MHD) model equations. Unlike traditional approaches that rely on satisfying the Ladyzhenskaya–Babuska–Brezzi (LBB) or inf-sup conditions, this work introduces a new stabilization term to ensure stability without the need for such classical conditions. This innovation not only simplifies the computational process by eliminating the requirement for additional parameters or higher-order derivatives but also enhances the efficiency and effectiveness of numerical approximate results and convergence. Moreover, the proposed technique offers a seamless integration into existing computational process, requiring only small modifications. Overall, this innovation presents a significant advancement in the field of non-linear MHD modeling, promising improved accuracy, stability, and computational efficiency for a wide range of utilizations.

The other part of the paper is organized as follows: The magnetohydrodynamic differential equations, as well as related notations and preliminary information, are presented in Section 2. The weak formulation and several well-known lemmas are given in Section 3. In Section 4, the linearized stabilized finite element formulation is provided together with the optimal convergence analysis and well-posedness derivation. To confirm the efficiency and precision of the method, we present numerical results in Section 5. Section 6 provides a short overview of the main conclusions derived from this research.

2. Magnetohydrodynamic Differential Equations and Preliminaries

2.1. Flowchart

The flowchart provides a systematic overview of the steps involved in solving the non-linear incompressible MHD model equations (Figure 1) using the lowest equal order mixed FEM, from problem formulation to result interpretation and documentation.

2.2. The Strong Form

In this work, a common version of the incompressible magnetohydrodynamics (MHD) equations in a complex domain $\mathfrak{D}\subset {\mathbb{R}}^d$, in which d = 2, 3, is addressed. These equations include the strong coupling between the Navier–Stokes equations and electromagnetics, giving rise to a highly nonlinear model. These nonlinear equations have been widely established and analyzed in various other studies, including [2,14,15,16,17].

Consider the hydrodynamic velocity $\overrightarrow{U}=(u_1,u_2,u_3)$, the hydrodynamic pressure p, the magnetic field $\overrightarrow{B}=(b_1,b_2,b_3)$ (i.e., the magnetic induction), and the magnetic pressure (Lagrange multiplier) r as unknowns. To obtain an approximate solution for these variables (refer to [16,18,19]), the following well-known partial differential equations will be utilized:

$$\begin{array}{ccc}\hfill -\frac{1}{R_e}\Delta \overrightarrow{U}+\overrightarrow{U}·\nabla \overrightarrow{U}+\nabla p+\Psi (\overrightarrow{B}\times curl\overrightarrow{B})& =& \overrightarrow{f},\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill \nabla ·\overrightarrow{U}& =& 0,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill \frac{1}{R_m}curl(curl\overrightarrow{B})+\nabla r-\Psi (curl(\overrightarrow{U}\times \overrightarrow{B}))& =& \overrightarrow{0},\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill \nabla ·\overrightarrow{B}& =& 0.\hfill \end{array}$$

(4)

In this context, $R_e$ refers to the hydrodynamic Reynolds number, $R_m$ to the magnetic Reynolds number, and Ψ to the coupling number. The external force term is denoted by the function $\overrightarrow{f}$. When working in two dimensions, the curl operator acting on a vector $\overrightarrow{B}=(b_1,b_2)$ can be defined as $\nabla \times \overrightarrow{B}=\frac{\mathsf{\partial }{b}_1}{\mathsf{\partial }y}-\frac{\mathsf{\partial }{b}_2}{\mathsf{\partial }x}$. On the other hand, the cross-product of two vectors can be expressed as $\overrightarrow{U}(u_1,u_2)\times \overrightarrow{B}(b_1,b_2)=u_1{b}_2-u_2{b}_1$. In the given formulation, r represents a scalar function denoting magnetic pressure, which can be determined by $\nabla \times r=(\frac{\mathsf{\partial }r}{\mathsf{\partial }y},-\frac{\mathsf{\partial }r}{\mathsf{\partial }x})$.

Remark 1. 

The MHD model equation includes the Navier–Stokes and Maxwell equations, where (1) illustrates linear momentum conservation, (2) describes continuity equation or mass conservation equation, (3) is the magnetic induction equation, and (4) confirms the isolated single magnetic poles (north or south).

The system works within a limited and connected full domain $\mathfrak{D}$ in $R^d$, and d = (2,3) represents dimensions of the domain. The basic homogeneous Dirichlet boundary conditions have been precisely defined for boundaries ($\mathsf{\partial }\mathfrak{D}$) [5,20].

$$\begin{array}{ccc}\hfill \overrightarrow{U}& =& 0,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill \overrightarrow{B}·\overrightarrow{n}& =& 0,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill curl\overrightarrow{B}\times \overrightarrow{n}& =& 0,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill r& =& 0,\hfill \end{array}$$

(8)

where $\overrightarrow{n}$ represents a normal vector on the domain $\mathfrak{D}$. The scalar functions p and r must have a mean value of zero over the entire domain.

2.3. Problem Linearization

Before proceeding with the finite element approximation, consider a linearized form of the magnetohydrodynamics (MHD) equations. The most straightforward method for this linearization is to use fixed-point handling of quadratic terms. Consider the model equation with known values for the velocity and magnetic field at the initial iteration n, indicated as ${\overrightarrow{U}}^n$ and ${\overrightarrow{B}}^n$. We aim to find these fields at the unknown iteration $n+1$. It is assumed that $\nabla ·{\overrightarrow{U}}^n=0$ and $\nabla ·{\overrightarrow{B}}^n=0$.

In the Navier–Stokes equations, the convective terms $\overrightarrow{U}·\nabla {\overrightarrow{U}}^{n+1}$ are non-linear. These non-linear terms create a coercive linearized issue. The remedy for the non-linear terms in the magnetic field is to replace $\overrightarrow{b}={\overrightarrow{B}}^n$ for coercivity. Therefore, generally, the substitutions $\overrightarrow{a}\equiv {\overrightarrow{U}}^n$, $\overrightarrow{U}\equiv {\overrightarrow{U}}^{n+1}$, $\overrightarrow{b}\equiv {\overrightarrow{B}}^n$, and $\overrightarrow{B}\equiv {\overrightarrow{B}}^{n+1}$ can be made initially (see [14,21], and the references cited therein).

The non-dimensional linear model equation is now provided by a set of differential equations to be solved in the bounded domain with given boundary conditions (see Equation (5)).

$$\begin{array}{ccc}\hfill -\frac{1}{R_e}\Delta \overrightarrow{U}+\overrightarrow{a}·\nabla \overrightarrow{U}+\nabla p+\Psi (\overrightarrow{b}\times curl\overrightarrow{B})& =& \overrightarrow{f},\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill \nabla ·\overrightarrow{U}& =& 0,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill \frac{1}{R_m}curl(curl\overrightarrow{B})+\nabla r-\Psi (curl(\overrightarrow{U}\times \overrightarrow{b}))& =& \overrightarrow{0},\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill \nabla ·\overrightarrow{B}& =& 0.\hfill \end{array}$$

(12)

2.4. Weak Formulation

For further mathematical setting, some definitions are recalled: for $k\in N$, the norm associated with Sobolev space $W^{k,p}(\mathfrak{D})$ by $\mid \mid ⨀\mid {\mid }_{W^{k,p}}$, with the special case $W^{k,2}(\mathfrak{D})$ being written as $H^k(\mathfrak{D})$ with the norm $\mid \mid ⨀\mid {\mid }_k$ and seminorm ${|⨀|}_k$ [22]. If $1\le p\le \infty$ and $k=0$, then $W_2^0(\mathfrak{D}).$ We denote $H_0^1(\mathfrak{D})$ as the subspace of $H^1(\mathfrak{D})$ of functions vanishing on $\mathsf{\partial }\mathfrak{D}$ denoted as $\mid \mid ·\mid {\mid }_{1,\mathfrak{D}}$.

The curl-norm is defined as $\Vert \overrightarrow{B}{\Vert }_{H(curl;\mathfrak{D})}={\left(\Vert \overrightarrow{B}{\Vert }_{L^2(\mathfrak{D})}^2+{\Vert \nabla ·\overrightarrow{B}\Vert }_{L^2(\mathfrak{D})}^2\right)}^{1/2}$. The space $H(curl;\mathfrak{D})$ is the space of vector field $\overrightarrow{c}\in L^2{(\mathfrak{D})}^2$ denoted by ${\Vert ⨀\Vert }_{curl}$. Furthermore, the norm $\overrightarrow{m}\mapsto {\Vert curl\overrightarrow{m}\Vert }_0$ is equivalent to the norm ${\Vert ⨀\Vert }_{curl}$; see, e.g., [23].

To introduce a weak formulation for the MHD model Equations (9)–(12), the following functional spaces are used in the analysis for the unknown variables (velocity, pressure of hydrodynamics, magnetic field, magnetic force) as $\mathcal{X}$, $\mathcal{Q}$, $\mathcal{W}$, $\mathcal{M}$, respectively:

$$\begin{array}{ccc}\hfill \mathrm{Velocity}\mathrm{Space}:\mathcal{X}:H_0^1(\mathfrak{D})& =& \{\overrightarrow{u}\in H^1{(\mathfrak{D})}^2:\overrightarrow{u}=\overrightarrow{0}on\mathsf{\partial }\mathfrak{D}\},\hfill \\ \hfill \mathrm{Magnetic}\mathrm{field}\mathrm{Space}:\mathcal{W}:H(curl;\mathfrak{D})& =& \{\overrightarrow{m}\in L^2{(\mathfrak{D})}^2:curl\overrightarrow{m}\in L^2{(\mathfrak{D})}^2,\overrightarrow{n}\times \overrightarrow{m}{|}_{\mathsf{\partial }\mathfrak{D}}=\overrightarrow{0}\},\hfill \\ \hfill \mathrm{Hydro.}\mathrm{pressure}\mathrm{Space}:\mathcal{Q}:L_0^2(\mathfrak{D})& =& \{q\in L^2(\mathfrak{D});{\int }_{\mathfrak{D}}qdx=0\},\hfill \\ \hfill \mathrm{Magnetic}\mathrm{pressure}\mathrm{Space}:\mathcal{M}:H_0^1(\mathfrak{D})& =& \{s\in H^1(\mathfrak{D}):s=0on\mathsf{\partial }\mathfrak{D}\}.\hfill \end{array}$$

By multiplying test functions $(\overrightarrow{u},q,\overrightarrow{m},s)$, integrating by parts, and utilizing boundary conditions, the weak/variational form of the above system (9)–(12) can be written [10,24] as

$$\begin{array}{ccc}\hfill \nu (\nabla \overrightarrow{U},\nabla \overrightarrow{u})+(\overrightarrow{a}·\nabla \overrightarrow{U},\overrightarrow{u})-(p,\nabla ·\overrightarrow{u})+\mathsf{\Psi }(\overrightarrow{b}\times curl\overrightarrow{B},\overrightarrow{u})& =& (\overrightarrow{f},\overrightarrow{u})\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill (q,\nabla ·\overrightarrow{U})& =& 0,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill {\nu }_m(curl\overrightarrow{B},curl\overrightarrow{m})-(r,\nabla ·\overrightarrow{m})-\mathsf{\Psi }(\overrightarrow{U}\times \overrightarrow{b},curl\overrightarrow{m})& =& \overrightarrow{0},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill (s,\nabla ·\overrightarrow{B})& =& 0.\hfill \end{array}$$

(16)

The parameters $\nu =\frac{1}{R_e}$ kinematic viscosity, ${\nu }_m=\frac{1}{R_m}$ denotes and magnetic diffusivity and $(·,·)$ inner product, respectively, while Ψ represents the coupling term. The artificial magnetic pressure r serves as a Lagrange multiplier in the induction equation, imposing the divergence-free limitation on the Maxwell equation in discrete conditions; nevertheless, in continuous situations, r is zero. The variables $\overrightarrow{U}$, $\overrightarrow{B}$, p, and r remain dimensionless. We might not utilize the curl formulation of the induction equation if the domain is sufficiently smooth, which is a common assumption in geophysics and astronomy. Therefore, the divergence of a vector function’s curl is always zero, the same as it is with a solenoidal vector field.

For further utilization, Equations (13)–(16) can be rewritten in a concise form to find the solutions $(\overrightarrow{U},p,\overrightarrow{B},r)\in \mathcal{X}\times \mathcal{Q}\times \mathcal{W}\times \mathcal{M}$, where

$$\begin{array}{c}\hfill \mathcal{N}((\overrightarrow{U},p,\overrightarrow{B},r),(\overrightarrow{u},q,\overrightarrow{m},s)=F(\overrightarrow{u},0,\overrightarrow{0},0)\end{array}$$

(17)

whereas [10]

$$\begin{array}{cc}\hfill \mathcal{N}((\overrightarrow{U},p,\overrightarrow{B},r),(\overrightarrow{u},q,\overrightarrow{m},s))=& \nu (\nabla \overrightarrow{U},\nabla \overrightarrow{u})+(\overrightarrow{a}·\nabla \overrightarrow{U},\overrightarrow{u})-(p,div\overrightarrow{u})\hfill \\ & +\mathsf{\Psi }(\overrightarrow{b}\times curl\overrightarrow{B},\overrightarrow{u})+(q,\nabla ·\overrightarrow{U})\hfill \\ & +{\nu }_m(curl\overrightarrow{B},curl\overrightarrow{m})-\mathsf{\Psi }(\overrightarrow{U}\times \overrightarrow{b},curl\overrightarrow{m})\hfill \\ & -(r,\nabla ·\overrightarrow{m})+(s,\nabla ·\overrightarrow{B}).\hfill \end{array}$$

(18)

Suppose $(\nabla ·\overrightarrow{a}=0)$ and $(\nabla ·\overrightarrow{b}=0)$ in the theoretical part of this work. A brief discussion is presented in the work of Codina [21] and resources cited therein about the system Equation (17) problem’s existence and uniqueness. It should be noted that, in the mixed FE method, regulation of the hydrodynamic velocity $\overrightarrow{U}$ in $H_0^1(\mathfrak{D})$ and of the magnetic field $\overrightarrow{B}$ in $H(curl,\mathfrak{D})$, as well as consideration of boundary conditions, are achievable (refer to [21] for further details). Moreover, ensuring the stability of the discrete scheme requires satisfaction of the important inf-sup criteria when substituting discrete subspaces with continuous function spaces. For instance, the following inf-sup conditions need to be fulfilled in order to control p and r [2].

$$\underset{q\in \mathcal{Q}}{inf}\underset{\overrightarrow{u}\in \mathcal{X}}{sup}\frac{(q,\nabla ·\overrightarrow{u})}{\Vert \overrightarrow{u}{\Vert }_1{\Vert q\Vert }_0}\ge {\beta }_1>0.$$

(19)

Here, ${\beta }_1$ is a positive constant dependent on a domain.

$$\underset{s\in \mathcal{M}}{inf}\underset{\overrightarrow{m}\in \mathcal{W}}{sup}\frac{(s,\nabla ·\overrightarrow{m})}{\Vert \overrightarrow{m}{\Vert }_{curl}{\Vert s\Vert }_1}\ge {\beta }_2>0.$$

(20)

The above two conditions (19) and (20) require strict choices regarding the FE spaces. This contribution’s purpose is to evaluate a stabilized FE formulation that does not have these kinds of requirements and enables the application of lowest-order FE pairings for all unknowns.

Simply, in this work, it is not our concern to discuss the inf-sup conditions in detail. More details can be seen in [4]. However, the immediate consequence of (19) and (20) will also apply to the following product form:

$$\underset{(q,s)\in \mathcal{Q}\times \mathcal{M}}{inf}\underset{(\overrightarrow{u},\overrightarrow{m})\in \mathcal{X}\times \mathcal{W}}{sup}\frac{\mathbb{Z}(\overrightarrow{u},\overrightarrow{m}:q,s)}{\Vert \overrightarrow{u},\overrightarrow{m}{\Vert }_{\mathcal{X}\times \mathcal{W}}{\Vert q,s\Vert }_{\mathcal{Q}\times \mathcal{M}}}\ge {\beta }_3>0.$$

(21)

Remark 2. 

The standard condition for pressure and velocity in the incompressible Navier–Stokes equations is (19), corresponding to the standard FE well-posedness in the continuous situation. Also, the second condition (20) is easily verified in the continuous case for $\overrightarrow{m}=\nabla s$, which gives an inf-sup = 1, which indicates ${\beta }_2\ge 1$. On the other hand, condition (20) is satisfied for random FE estimates of the magnetic field and the magnetic pressure for equal elements. As a result, it is not feasible to take $\overrightarrow{m}=\nabla s$ in the discrete formulation. We modify the variation formulation and add a similar additional stabilizing component to the p and r in order to decrease the requirement for these two inf-sup demands and ensure the well-posedness of the MHD model. We will utilize the same method in our contribution.

2.5. Solenoidal Function Spaces

For the convenience of analysis, it is important to recall some solenoidal function spaces in the this section. This idea has already been discussed in the second section of [23]. Spaces

$$H(\mathfrak{D})=H(curl;\mathfrak{D})\bigcap H_0(\nabla ·;\mathfrak{D}),$$

are fitted with the norm

$$\Vert \overrightarrow{m}{\Vert }_1^2=\Vert \overrightarrow{m}{\Vert }_0^2+\Vert \overrightarrow{m}{\Vert }_{curl}^2+{\Vert \nabla ·\overrightarrow{m}\Vert }_0^2.$$

Next, rewrite $\mathcal{W}=H(\mathfrak{D})$ with the norm added:

$$\begin{array}{ccc}\hfill \Vert curl\overrightarrow{m}{\Vert }_{curl}^2& =& (\Vert \overrightarrow{m}{\Vert }_0^2+\Vert curl\overrightarrow{m}{{\Vert }_0^2)}^{1/2},\hfill \end{array}$$

where $\mathcal{W}$ is the Poincaré–Friedrichs inequality:

$$\begin{array}{ccc}\hfill \Vert curl\overrightarrow{m}{\Vert }_0^2& \ge & C\Vert \overrightarrow{m}{\Vert }_0^2,\forall \overrightarrow{m}\in \mathcal{W}.\hfill \end{array}$$

(22)

Equation (22) indicates that variables throughout $\mathcal{W}$ are uniquely established by their rotation.

3. Finite Element Spatial Discretization

The MHD model equation with spatial discretization (13)–(16) will be employed in this section. The notations and norms from [12,23] are utilized to introduce the mixed finite element (FE) strategy. Several fundamental vector function spaces are given by

$$\begin{array}{ccc}\hfill {\mathcal{X}}_h& :=& \{\overrightarrow{u}\in \mathcal{X}\cap C^0{(\tilde{\mathfrak{D}})}^2;{\overrightarrow{u}}_{|E}\in P_1{(E)}^2,\forall E\in T_h\},\hfill \\ \hfill {\mathcal{Q}}_h& :=& \{q\in \mathcal{Q}\cap C^0(\tilde{\mathfrak{D}});q_{|E}\in P_1(E);\forall E\in T_h\},\hfill \\ \hfill {\mathcal{W}}_h& :=& \{\overrightarrow{m}\in \mathcal{W}\cap C^0{(\tilde{\mathfrak{D}})}^2;{\overrightarrow{m}}_{|E}\in P_1{(E)}^2,\forall E\in T_h\},\hfill \\ \hfill {\mathcal{M}}_h& :=& \{s\in \mathcal{M}\cap C^0(\tilde{\mathfrak{D}});s_{|E}\in P_1(E);\forall E\in T_h\}.\hfill \end{array}$$

The discrete variational formulation of the model equation can be written as

Problem $(D)$: Find $({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h)$$\in ({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{W}}_h\times {\mathcal{M}}_h)$ such that

$$\begin{array}{ccc}\hfill \nu (\nabla {\overrightarrow{U}}_h,\nabla {\overrightarrow{u}}_h)+(\overrightarrow{a}·\nabla {\overrightarrow{U}}_h,{\overrightarrow{u}}_h)-(p_h,\nabla ·{\overrightarrow{u}}_h)+\mathsf{\Psi }(\overrightarrow{b}\times curl{\overrightarrow{B}}_h,{\overrightarrow{u}}_h)& =& (\overrightarrow{f},{\overrightarrow{u}}_h),\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill (q_h,\nabla ·{\overrightarrow{U}}_h)& =& 0,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill {\nu }_m(curl{\overrightarrow{B}}_h,curl{\overrightarrow{m}}_h)+(\nabla r_h,{\overrightarrow{m}}_h)-\mathsf{\Psi }({\overrightarrow{U}}_h\times \overrightarrow{b},curl{\overrightarrow{m}}_h)& =& 0.\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill -(\nabla s_h,{\overrightarrow{B}}_h)& =& 0.\hfill \end{array}$$

(26)

The system of Equations (23)–(26) can be rewritten in simplest form as on $({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{W}}_h\times {\mathcal{M}}_h)$ by

$$\begin{array}{ccc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h))& =& F({\overrightarrow{u}}_h),\hfill \end{array}$$

(27)

where the bilinear form $N(·,·,·,·)$ is defined as:

$$\begin{array}{cc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h))=& \nu (\nabla {\overrightarrow{U}}_h,\nabla {\overrightarrow{u}}_h)+(\overrightarrow{a}·\nabla {\overrightarrow{U}}_h,{\overrightarrow{u}}_h)-(p_h,\nabla ·{\overrightarrow{u}}_h)\hfill \\ & +\mathsf{\Psi }(\overrightarrow{b}\times curl{\overrightarrow{B}}_h,{\overrightarrow{u}}_h)+(q_h,\nabla ·{\overrightarrow{U}}_h)\hfill \\ & +{\nu }_m(curl{\overrightarrow{B}}_h,curl{\overrightarrow{m}}_h)-\mathsf{\Psi }({\overrightarrow{U}}_h\times \overrightarrow{b},curl{\overrightarrow{m}}_h)\hfill \\ & +(\nabla r_h,{\overrightarrow{m}}_h)-(\nabla s_h,{\overrightarrow{B}}_h).\hfill \end{array}$$

Similarly, the linear form of the Equation (27) is

$$F({\overrightarrow{u}}_h)=f(\overrightarrow{u},0,\overrightarrow{0},0).$$

The variational problem raised by Equation (27) is a problem of a mixed type and has a number of parallels to the mixed variational method used to deal with the Navier–Stokes equations (Brezzi and Fortin [25]). Hence, in order to obtain a stable numerical solution, it is necessary to be consistent against the inf-sup stability requirement, also known as the LBB condition. It is practically essential to use different estimates for the two variables in order to meet this stability requirement. Using components that satisfy the LBB requirements in the context of three-dimensional problems may result in higher computational expenses in terms of memory utilization and execution time.

In addition, dealing with mixed formulations involving three dimensions can be challenging. By eliminating the LBB condition [26], a stabilized finite element (FE) approach for the system (27) can be developed. The pressure is inconsistent and does not satisfy the inf-sup requirements due to the choice of lowest equal order FE pairings. Consequently, a stabilization term is needed to overcome the deficiency of the inf-sup condition.

The operators $G_1(·,·)$ and $G_2(·,·)$ are defined here, as covered in the literature [27,28]. $\mathbb{I}$ is the identity operator, $\mathsf{\Theta }:L^2(\mathfrak{D})\to R_0$ is a local pressure projection operator, and $R_0\subset L^2(\mathfrak{D})$ is the finite element space of piecewise constants on the established domain, satisfying the conditions described in [13]. Based on these representations, the following relations hold:

$$\begin{array}{ccc}\hfill (p,q)& =& (\mathsf{\Theta }p,q)\forall q\in L^2(\mathfrak{D}),q\in R_0,\hfill \\ \hfill {\left|\right|\mathsf{\Theta }p\left|\right|}_0& \le & C^{\ast }{\left|\right|p\left|\right|}_0\forall p\in L^2(\mathfrak{D}),\hfill \\ \hfill {\left|\right|(\mathbb{I}-\mathsf{\Theta })p\left|\right|}_0& \le & C^{\ast }{h\left|\right|p\left|\right|}_1\forall p\in Q.\hfill \end{array}$$

The discrete formulation of the local pressure estimates is listed below:

$$G(p_h,q_h)=(p_h-\mathsf{\Theta }p,q_h-\mathsf{\Theta }q).$$

To our knowledge, this technique has not yet been presented in the MHD literature. Well-known results from the literature, including [13,27,29], were adapted to establish the stability of the method. In investigating and establishing inequalities, global positive constants denoted by C, with either superscripts or subscripts, are considered, depending solely on the domain $\omega ,\mathsf{\Psi },f$ and independent of the mesh size h. The meaning of these constants may vary depending on the context.

Lemma 1 

([12,13]). Utilizing the previously defined spaces for pressure and velocity, ${\mathcal{X}}_h$ and ${\mathcal{Q}}_h$, respectively, will be considered. Following that, two positive constants, $C_1$ and $C_2$, appear, ensuring that:

$$\underset{{\overrightarrow{u}}_h\in {\mathcal{X}}_h}{sup}\frac{{\int }_{\mathfrak{D}}{p}_h\nabla ·{\overrightarrow{u}}_{h}d\mathfrak{D}}{\Vert {\overrightarrow{u}}_h{\Vert }_1}\ge C_1\Vert p_h{\Vert }_0-C_{2}h{\Vert \nabla p_h\Vert }_0.$$

(28)

Proof. 

See [13] for the complete proof. □

Lemma 2. 

A positive nonzero constant $C_3$ appears such that

$$\begin{array}{ccc}\hfill C_{3}h{\Vert \nabla p_h\Vert }_0& \ge & \Vert p_h-\mathsf{\Theta }{p}_h{\Vert }_0.\hfill \end{array}$$

(29)

Proof. 

Thanks to [13]. □

As for the magnetic field and the solenoidal field, we have

Lemma 3. 

Suppose that the spaces defined ${\mathcal{W}}_h$ and ${\mathcal{M}}_h$. Then, positive constants $C_4$ and $C_5$ hold in a way that

$$\underset{{\overrightarrow{m}}_h\in {\mathcal{W}}_h}{sup}\frac{{\int }_{\mathfrak{D}}{r}_h\nabla ·{\overrightarrow{m}}_{h}d\mathfrak{D}}{\Vert {\overrightarrow{m}}_h{\Vert }_{curl}}\ge C_4\Vert r_h{\Vert }_1-C_{5}h{\Vert \nabla r_h\Vert }_0,\forall r_h\in {\mathcal{M}}_h.$$

(30)

Proof. 

By using the definition of ${\mathcal{W}}_h$, every $r_h\in {\mathcal{M}}_h$ also belongs to $L^2(\mathfrak{D})$. So as a result, there exists ${\overrightarrow{w}}_0\in H_{curl}(\mathfrak{D})$ such that

$${\int }_{\mathfrak{D}}{r}_h\nabla ·{\overrightarrow{w}}_{0,h}d\mathfrak{D}\ge C^6\Vert r_h{\Vert }_1{\Vert {\overrightarrow{w}}_0\Vert }_{curl}.$$

(31)

Let ${\overrightarrow{w}}_{0,h}$ denote the interpolant of $\overrightarrow{w}$ out of ${\mathcal{W}}_h$, then the following inequality holds:

$$\Vert {\overrightarrow{w}}_0-{\overrightarrow{w}}_{0,h}{\Vert }_0\le C_{7}h\Vert {\overrightarrow{w}}_0{\Vert }_{1}and\Vert {\overrightarrow{w}}_{0,h}{,\Vert }_{curl}\le C_8{\Vert {\overrightarrow{w}}_0\Vert }_1.$$

(32)

Hence, by using (22) and the incompressibility condition, we obtain

$$\begin{array}{cc}\hfill \Vert {\overrightarrow{w}}_0{\Vert }_1& =\Vert \nabla ·{\overrightarrow{w}}_0{\Vert }_0+\Vert {\overrightarrow{w}}_0{\Vert }_{curl}+{\Vert {\overrightarrow{w}}_0\Vert }_0\hfill \\ & \le C_9{\Vert {\overrightarrow{w}}_0\Vert }_{curl}.\hfill \end{array}$$

(33)

By using (31), (32), (33) and the fact that ${\mathcal{M}}_h$ are piecewise continuous functions,

$$\begin{array}{cc}\hfill \frac{\mid {\int }_{\mathfrak{D}}{r}_h\nabla ·{\overrightarrow{w}}_{0,h}d\mathfrak{D}\mid }{\Vert {\overrightarrow{w}}_h{\Vert }_{curl}}& \ge \frac{\mid {\int }_{\mathfrak{D}}{r}_h\nabla ·{\overrightarrow{w}}_{0,h}d\mathfrak{D}\mid }{C_{1}0{\Vert {\overrightarrow{w}}_0\Vert }_{curl}}\hfill \\ & =\frac{\mid {\int }_{\mathfrak{D}}{r}_h\nabla ·({\overrightarrow{w}}_{0,h}-{\overrightarrow{w}}_0)d\mathfrak{D}+{\int }_{\mathfrak{D}}{r}_h\nabla ·{\overrightarrow{w}}_{0}d\mathfrak{D}\mid }{C_{1}1{\Vert {\overrightarrow{w}}_0\Vert }_{curl}}\hfill \\ & \ge \frac{{\int }_{\mathfrak{D}}{r}_h\nabla ·{\overrightarrow{w}}_{0}d\mathfrak{D}}{C\Vert {\overrightarrow{w}}_0{\Vert }_{curl}}+\frac{\mid {\int }_{\mathfrak{D}}{r}_h\nabla ·({\overrightarrow{w}}_{0,h}-{\overrightarrow{w}}_0)d\mathfrak{D}\mid }{C\Vert {\overrightarrow{w}}_0{\Vert }_{curl}}\hfill \\ & \ge \frac{C^{\ast }}{C}{\Vert r_h\Vert }_1-\frac{\mid {\int }_{\mathfrak{D}}\nabla r_h·({\overrightarrow{w}}_{0,h}-{\overrightarrow{w}}_0)d\mathfrak{D}\mid }{C\Vert {\overrightarrow{w}}_0{\Vert }_{curl}}\hfill \\ & \ge C_4{\Vert r_h\Vert }_1-\frac{\Vert \nabla r_h{\Vert }_0{\Vert {\overrightarrow{w}}_0-{\overrightarrow{w}}_h\Vert }_0}{C\Vert {\overrightarrow{w}}_0{\Vert }_{curl}}\hfill \\ & \ge C_4{\Vert r_h\Vert }_1-\frac{\Vert \nabla r_h{\Vert }_{0}Ch{\Vert {\overrightarrow{w}}_0\Vert }_1}{C\Vert {\overrightarrow{w}}_0{\Vert }_{curl}},\hfill \\ & \ge C_4\Vert r_h{\Vert }_1-C_{5}h{\Vert \nabla r_h\Vert }_0.\hfill \end{array}$$

(34)

Then, the inequality holds as

$$\begin{array}{ccc}\hfill \underset{{\overrightarrow{m}}_h\in {\mathcal{W}}_h}{sup}\frac{{\int }_{\mathfrak{D}}{r}_h\nabla ·{\overrightarrow{m}}_{h}d\mathfrak{D}}{\Vert {\overrightarrow{m}}_h{\Vert }_{curl}}& \ge & \frac{\mid {\int }_{\mathfrak{D}}{r}_h\nabla ·{\overrightarrow{w}}_{0,h}d\mathfrak{D}\mid }{\Vert {\overrightarrow{w}}_{0,h}{\Vert }_{curl}}.\hfill \end{array}$$

(35)

□

Lemma 4. 

There is a positive, non-zero constant $C^7$ such that

$$\begin{array}{ccc}\hfill C^{7}h{\Vert \nabla r_h\Vert }_0& \ge & \Vert r_h-\mathsf{\Theta }{r}_h{\Vert }_0.\hfill \end{array}$$

(36)

Proof. 

Thanks to [13]. □

The Finite Element Stabilized Method

In the following subsection, a stable approximation approach for problem (27) using finite elements is demonstrated. We initiate by describing the challenge as a system of linear equations for convection and diffusion. Our finite element approach will be explained within this larger system.

Several reasons exist for addressing this issue. Primarily, it is aimed to circumvent the necessity of satisfying inf-sup or (LBB) conditions (19) and (20), which would entail numerous interpolations (high-level interpolation polynomials) for the relevant variables. Challenges are simplified by assuming that all unknowns will be uniformly and continuously interpolated (linear polynomials $p1-p1-p1-p1$). The second objective of stabilization is to provide reliable error estimates, particularly in convection-dominated situations such as those encountered in the magnetic field equation and the Navier–Stokes equation. Lastly, appropriate management of the complex relationship between the magnetic and hydrodynamic challenges is sought. These objectives will be clearly demonstrated in the error estimation.

Problem$(D2)$: Find $({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h)$$\in ({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{W}}_h\times {\mathcal{M}}_h)$ such that

$$\begin{array}{ccc}\hfill \nu (\nabla {\overrightarrow{U}}_h,\nabla {\overrightarrow{u}}_h)+(\overrightarrow{a}\nabla {\overrightarrow{U}}_h,{\overrightarrow{u}}_h)-(p_h,\nabla ·{\overrightarrow{u}}_h)+\mathsf{\Psi }(\overrightarrow{b}\times curl{\overrightarrow{B}}_h,{\overrightarrow{u}}_h)& =& (\overrightarrow{f},{\overrightarrow{u}}_h),\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill (q_h,\nabla ·{\overrightarrow{U}}_h)+G_1(p_h,q_h)& =& 0,\hfill \end{array}$$

(38)

$$\begin{array}{ccc}\hfill {\nu }_m(curl{\overrightarrow{B}}_h,curl{\overrightarrow{m}}_h)+(\nabla r_h,{\overrightarrow{m}}_h)-\mathsf{\Psi }({\overrightarrow{U}}_h\times \overrightarrow{b},curl{\overrightarrow{m}}_h)& =& 0.\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill (\nabla s_h,{\overrightarrow{B}}_h)+G_2(r_h,s_h)& =& 0.\hfill \end{array}$$

(40)

The discrete systems of equations can be equivalently modified in the following form:

$$\begin{array}{ccc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h))& =& F({\overrightarrow{u}}_h),\hfill \end{array}$$

(41)

where the bilinear form $\mathcal{N}(·,·,·,·)$ is defined as

$$\begin{array}{cc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h),({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h))=& \nu (\nabla {\overrightarrow{U}}_h,\nabla {\overrightarrow{u}}_h)+(\overrightarrow{a}·\nabla {\overrightarrow{U}}_h,{\overrightarrow{u}}_h)-(p_h,\nabla ·{\overrightarrow{u}}_h)\hfill \\ & +\mathsf{\Psi }(\overrightarrow{b}\times curl{\overrightarrow{B}}_h,{\overrightarrow{u}}_h)+(q_h,\nabla ·{\overrightarrow{U}}_h)\hfill \\ & +{\nu }_m(curl{\overrightarrow{B}}_h,curl{\overrightarrow{m}}_h)-\mathsf{\Psi }({\overrightarrow{U}}_h\times \overrightarrow{b},curl{\overrightarrow{m}}_h)+G_1(p_h,q_h)\hfill \\ & +(\nabla r_h,{\overrightarrow{m}}_h)-(\nabla s_h,{\overrightarrow{B}}_h)+G_2(r_h,s_h)\hfill \end{array}$$

(42)

and the linear functional is $F({\overrightarrow{u}}_h)=(\overrightarrow{f},\overrightarrow{u},0,0,0)$. This section concludes with several approximation facts and inequalities. Let ${\tilde{\overrightarrow{U}}}_h\in {\mathcal{X}}_h$ be defined as the interpolant of $\overrightarrow{U}$ in $\mathcal{X}$, ${\tilde{p}}_h\in Q_h$ be the projected orthogonal of $p\in \mathcal{Q}$, and ${\tilde{\overrightarrow{B}}}_h\in {\mathcal{W}}_h$ be the interpolant of ${\overrightarrow{B}}_h\in \mathcal{W}$, projection of $r\in \mathcal{M}$ orthogonally. The following results are satisfied:

$$\begin{array}{ccc}\hfill \Vert \overrightarrow{U}-{\tilde{\overrightarrow{U}}}_h{\Vert }_0+h{\Vert \nabla (\overrightarrow{U}-{\tilde{\overrightarrow{U}}}_h)\Vert }_0& \le & C^8{h}^2{\Vert \overrightarrow{U}\Vert }_2,\hfill \end{array}$$

(43)

$$\begin{array}{ccc}\hfill \Vert \overrightarrow{B}-{\tilde{\overrightarrow{B}}}_h{\Vert }_{curl}+h{\Vert curl(\overrightarrow{B}-{\tilde{\overrightarrow{B}}}_h)\Vert }_0& \le & Ch\Vert \overrightarrow{B}{\Vert }_1,\hfill \end{array}$$

(44)

$$\begin{array}{ccc}\hfill \Vert p-{\tilde{p}}_h{\Vert }_0& \le & C^{8}h{\Vert p\Vert }_1,\hfill \end{array}$$

(45)

$$\begin{array}{ccc}\hfill \Vert r-{\tilde{r}}_h{\Vert }_1& \le & C^8{h}^2{\Vert r\Vert }_2.\hfill \end{array}$$

(46)

Moreover, the inverse inequality is recalled as

$$\begin{array}{ccc}\hfill \Vert \nabla {\overrightarrow{u}}_h{\Vert }_0& \le & C{h}^{-1}{\Vert {\overrightarrow{u}}_h\Vert }_0,{\overrightarrow{u}}_h\in {\mathcal{X}}_h.\hfill \end{array}$$

(47)

$$\begin{array}{ccc}\hfill \Vert curl{\overrightarrow{m}}_h{\Vert }_0& \le & C{h}^{-1}{\Vert {\overrightarrow{m}}_h\Vert }_{curl},{\overrightarrow{m}}_h\in {\mathcal{W}}_h.\hfill \end{array}$$

(48)

4. Existence, Uniqueness, and Error Bound

This section investigates the problem’s existence and uniqueness. Before embarking on the study of the scheme’s well-posedness, we describe expected norms, including the Poincaré–Friedrichs inequality.

$$\begin{array}{ccc}\hfill \Vert {\overrightarrow{u}}_h{\Vert }_0& \le & C_P{\Vert \nabla {\overrightarrow{u}}_h\Vert }_0\forall {\overrightarrow{u}}_h\in {\mathcal{X}}_h,\hfill \end{array}$$

(49)

$$\begin{array}{ccc}\hfill \Vert {\overrightarrow{m}}_h{\Vert }_{curl}& \le & C_P{\Vert curl{\overrightarrow{m}}_h\Vert }_0\forall {\overrightarrow{m}}_h\in {\mathcal{W}}_h,\hfill \end{array}$$

(50)

with a positve constant $C_P$ [[30], Proposition 7.4 and Corollary 5.51 of [31]]. For all vectors $\overrightarrow{x},\overrightarrow{y}\in R^2$, there is $\mid \overrightarrow{x}·\overrightarrow{y}\mid \le \Vert \overrightarrow{x}\Vert \Vert \overrightarrow{y}\Vert$ as well as $\Vert \overrightarrow{x}\times \overrightarrow{y}\Vert \le \Vert \overrightarrow{x}\Vert \Vert \overrightarrow{y}\Vert$.

Theorem 1. 

Let us consider if $\overrightarrow{f}\in {\overrightarrow{L}}^2(\mathfrak{D})$, a unique solution holds for the $({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h)\in ({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{W}}_h\times {\mathcal{M}}_h)$ satisfying (41).

Proof. 

$$\begin{array}{cc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h))=& F({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)\hfill \\ \hfill & \forall ({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)\in ({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{W}}_h\times {\mathcal{M}}_h).\hfill \end{array}$$

(51)

A linear function $F(·)$ on Equation (51) can be written as:

$$F({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)=(\overrightarrow{f},{\overrightarrow{u}}_h).$$

And to prove the continuity of the linear operator $F(·)$, the Cauchy–Schwarz and Poincaré inequality can be utilized as:

$$\begin{array}{cc}\hfill \mid F({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)& \mid \le \Vert \overrightarrow{f}{\Vert }_0{\Vert {\overrightarrow{u}}_h\Vert }_0\hfill \\ \hfill & \le \Vert \overrightarrow{f}{\Vert }_0\mid \mid \mid ({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)\mid \mid {\mid }_{({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{W}}_h\times {\mathcal{M}}_h)}.\hfill \end{array}$$

(52)

where $\mid \mid \mid ({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)\mid \mid {\mid }_{({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{W}}_h\times {\mathcal{M}}_h)}=(\Vert {\overrightarrow{u}}_h{\Vert }_1^2+\Vert q_h{\Vert }_0^2+\Vert {\overrightarrow{m}}_h{\Vert }_{curl}^2+\Vert s_h{{\Vert }_1^2)}^{\frac{1}{2}}.$ The bilinear operator $\mathcal{N}$ used in (51) is continuous over (${\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{W}}_h\times {\mathcal{M}}_h$). Hence, to prove the continuity of $\mathcal{N}$ inequalities of Cauchy–Schwarz and Poincaré are utilized as:

$$\begin{array}{cc}\hfill \mathcal{N}& (({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h))\hfill \\ & \le \nu \Vert \nabla {\overrightarrow{U}}_h{\Vert }_0\Vert \nabla {\overrightarrow{u}}_h{\Vert }_0+\Vert \overrightarrow{a}{\Vert }_0\Vert \nabla {\overrightarrow{U}}_h{\Vert }_0\Vert {\overrightarrow{u}}_h{\Vert }_0-\Vert p_h{\Vert }_0{\Vert \nabla ·{\overrightarrow{u}}_h\Vert }_0\hfill \\ & +\Psi \Vert \overrightarrow{b}{\Vert }_0\Vert curl{\overrightarrow{B}}_h{\Vert }_0\Vert {\overrightarrow{u}}_h{\Vert }_0+\Vert q_h{\Vert }_0{\Vert \nabla ·{\overrightarrow{U}}_h\Vert }_0\hfill \\ & +{\nu }_m\Vert curl{\overrightarrow{B}}_h{\Vert }_0\Vert curl{\overrightarrow{m}}_h{\Vert }_0-\Vert {\overrightarrow{U}}_h\times \overrightarrow{b}{\Vert }_0\Vert curl{\overrightarrow{m}}_h{\Vert }_0+\Vert p_h{\Vert }_0{\Vert q_h\Vert }_0\hfill \\ & +\Vert \nabla r_h{\Vert }_0\Vert {\overrightarrow{m}}_h{\Vert }_0-\Vert \nabla s_h{\Vert }_0\Vert {\overrightarrow{B}}_h{\Vert }_0+\Vert r_h{\Vert }_0{\Vert s_h\Vert }_0\hfill \\ & \le \nu \Vert {\overrightarrow{U}}_h{\Vert }_1\Vert {\overrightarrow{u}}_h{\Vert }_1+\Vert \overrightarrow{a}{\Vert }_{\infty }\Vert {\overrightarrow{U}}_h{\Vert }_1{C}_P\Vert {\overrightarrow{u}}_h{\Vert }_1-\Vert p_h{\Vert }_0{\Vert \nabla ·{\overrightarrow{u}}_h\Vert }_0\hfill \\ & +\Psi \Vert \overrightarrow{b}{\Vert }_{\infty }\Vert {\overrightarrow{B}}_h{\Vert }_{curl}{C}_P\Vert {\overrightarrow{u}}_h{\Vert }_1+\Vert q_h{\Vert }_0{\Vert \nabla ·{\overrightarrow{U}}_h\Vert }_0\hfill \\ & +{\nu }_m\Vert {\overrightarrow{B}}_h{\Vert }_{curl}\Vert {\overrightarrow{m}}_h{\Vert }_{curl}-\Vert \overrightarrow{b}{\Vert }_{\infty }{C}_P\Vert {\overrightarrow{U}}_h{\Vert }_1\Vert {\overrightarrow{m}}_h{\Vert }_{curl}+\Vert p_h{\Vert }_0{\Vert q_h\Vert }_0\hfill \\ & +\Vert r_h{\Vert }_1{C}_P\Vert {\overrightarrow{m}}_h{\Vert }_{curl}-\Vert s_h{\Vert }_1{C}_P\Vert {\overrightarrow{B}}_h{\Vert }_{curl}+C_P\Vert r_h{\Vert }_1{\Vert s_h\Vert }_1\hfill \\ & \le C\mid \mid \mid ({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h)\mid \mid {\mid }_{({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{W}}_h\times {\mathcal{M}}_h)}\mid \mid \mid ({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)\mid \mid {\mid }_{({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{S}}_h\times {\mathcal{M}}_h).}\hfill \end{array}$$

(53)

Here, C is a positive constant depending on the domain, not necessarily the same at every place. It is noticeable that $\overrightarrow{a}$ and $\overrightarrow{b}$ are constants, following the same motivation from Codina et al. [21,32]. □

Next, the stability of the method will be demonstrated by utilizing theorems to establish that the bilinear form is coercive.

Theorem 2. 

There is a constant Γ such that $((\overrightarrow{U_h},p_h,\overrightarrow{B_h},r_h),(\overrightarrow{u_h},q_h,\overrightarrow{m_h},s_h))\in {\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{W}}_h\times {\mathcal{M}}_h$, the following inequality holds:

$$\begin{array}{c}\hfill \underset{({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)}{sup}\frac{\mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h))}{\mid \Vert {\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h\mid \Vert }\ge \mathsf{\Gamma }\mid \Vert ({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h)\mid \Vert \end{array}$$

(54)

Proof. 

To establish weak coercivity, the subsequent information is critical. A positive constant exists for every $p_h\in {\mathcal{Q}}_h\subset \mathcal{Q}$ and $r_h\in {\mathcal{M}}_h\subset \mathcal{M}$. $\mathsf{\Gamma }$ represents a constant (different for various sites) $\overrightarrow{\varphi }\in \mathcal{X}$ and ${\overrightarrow{\varphi }}_0\in \mathcal{W}$, such that for hydrodynamic pressure, consider:

$$(\nabla ·\overrightarrow{\varphi },p_h)=\Vert p_h{\Vert }_0^2,\Vert \overrightarrow{\varphi }{\Vert }_1\le \mathsf{\Gamma }{\Vert p_h\Vert }_0.$$

For the magneto pressure, we have

$$(\nabla ·{\overrightarrow{\varphi }}_0,r_h)=\Vert r_h{\Vert }_1^2,\Vert {\overrightarrow{\varphi }}_0{\Vert }_{curl}\le \mathsf{\Gamma }{\Vert r_h\Vert }_1.$$

Selecting a number in the FE approximation for normalization ${\overrightarrow{\varphi }}_h\in {\mathcal{X}}_h$ of $\overrightarrow{\varphi }$([11,12,13,28]), assume

$$\Vert {\overrightarrow{\varphi }}_h{\Vert }_1\le {\mathsf{\Gamma }}_0\Vert p_h{\Vert }_0,\Vert {\overrightarrow{\varphi }}_{0,h}{\Vert }_{curl}\le {\mathsf{\Gamma }}_0{\Vert r_h\Vert }_1.$$

(55)

Thanks to [13]

$${\int }_{\mathfrak{D}}{p}_h\nabla ·{\overrightarrow{\varphi }}_{h}d\mathfrak{D}\ge C^1\Vert p_h{\Vert }_0^2-C^2\Vert (\mathbb{I}-\mathsf{\Theta })p_h{\Vert }_0{\Vert p_h\Vert }_0.$$

(56)

The weak coercivity can be estimated by substitution of the operators in (51).

$$({\overrightarrow{u}}_h={\overrightarrow{U}}_h-\xi {\overrightarrow{\varphi }}_h,q_h=p_h,{\overrightarrow{m}}_h={\overrightarrow{B}}_h-\mu {\overrightarrow{\varphi }}_{0,h},s_h=r_h)$$

in which $\xi ,\mu \in \mathbb{R}$, corresponds to the bilinear form of $\mathcal{N}$:

$$\begin{array}{cc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),(& {\overrightarrow{U}}_h-\xi {\overrightarrow{\varphi }}_h,p_h,{\overrightarrow{B}}_h-\mu {\overrightarrow{\varphi }}_{0,h},r_h))\hfill \\ & =\mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h))\hfill \\ & +\mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),(-\xi {\overrightarrow{\varphi }}_h,0,0,0))\hfill \\ & +\mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),(0,0,-\mu {\overrightarrow{\varphi }}_{0,h},0)).\hfill \end{array}$$

(57)

Equation (57) can be represented by three extra terms, each of which can be confined as follows:

First term of (57).

Thanks to (42), we have

$$\begin{array}{cc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h))& \ge \nu \Vert \nabla {\overrightarrow{U}}_h{\Vert }_0^2+{\nu }_m\Vert curl{\overrightarrow{B}}_h{\Vert }_0^2+\Vert (\mathbb{I}-\mathsf{\Theta })p_h{\Vert }_0^2+{\Vert (\mathbb{I}-\mathsf{\Theta })r_h\Vert }_0^2.\hfill \end{array}$$

Second term of (57).

$$\begin{array}{cc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),(-\xi {\overrightarrow{\varphi }}_h,0,0,0))& =-\xi \nu (\nabla {\overrightarrow{U}}_h,\nabla {\overrightarrow{\varphi }}_h)-(\overrightarrow{a}·\nabla \overrightarrow{U},\xi {\overrightarrow{\varphi }}_h)-(\overrightarrow{b}\times curl{\overrightarrow{B}}_h,\xi {\overrightarrow{\varphi }}_h)\\ & +\xi (p_h,\nabla ·{\overrightarrow{\varphi }}_h).\end{array}$$

(58)

Equation (58) can be further estimated by applying Young’s inequality with Equations (43), (55), (56) giving:

$$\begin{array}{cc}\hfill \mid -\xi \nu (\nabla ({\overrightarrow{U}}_h),\nabla ({\overrightarrow{\varphi }}_h))\mid & \le \xi \nu \Vert \nabla {\overrightarrow{U}}_h{\Vert }_0{\Vert \nabla {\overrightarrow{\varphi }}_h\Vert }_0\hfill \\ & \le \xi \nu \Vert \nabla {\overrightarrow{U}}_h{\Vert }_0{\mathsf{\Gamma }}_0{\Vert p_h\Vert }_0\hfill \\ & \le \frac{\xi \nu {\mathsf{\Gamma }}_0^2}{{ϵ}_1}\Vert \nabla {\overrightarrow{U}}_h{\Vert }_0^2+\xi \nu {ϵ}_1{\Vert p_h\Vert }_0^2,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mid -(\overrightarrow{a}·\nabla \overrightarrow{U},\xi {\overrightarrow{\varphi }}_h)\mid & \le \xi \mid \overrightarrow{a}\mid \Vert \nabla {\overrightarrow{U}}_h{\Vert }_0{\Vert {\overrightarrow{\varphi }}_h\Vert }_0\hfill \\ & \le \xi \Vert \nabla {\overrightarrow{U}}_h{\Vert }_0{\mathsf{\Gamma }}_0{\Vert p_h\Vert }_0\hfill \\ & \le \frac{\xi {\mathsf{\Gamma }}_0^2}{{ϵ}_2}\Vert \nabla {\overrightarrow{U}}_h{\Vert }_0^2+\xi {ϵ}_2{\Vert p_h\Vert }_0^2,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mid -(\overrightarrow{b}\times curl{\overrightarrow{B}}_h,\xi {\overrightarrow{\varphi }}_h)\mid & \le \xi \mid \overrightarrow{b}\mid \Vert curl{\overrightarrow{B}}_h{\Vert }_0{\Vert {\overrightarrow{\varphi }}_h\Vert }_0\hfill \\ & \le \xi \Vert curl{\overrightarrow{B}}_h{\Vert }_0{\mathsf{\Gamma }}_0{\Vert p_h\Vert }_0\hfill \\ & \le \frac{\xi {\mathsf{\Gamma }}_0^2}{{ϵ}_3}\Vert curl{\overrightarrow{B}}_h{\Vert }_0^2+\xi {ϵ}_3{\Vert p_h\Vert }_0^2,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \xi (p_h,\nabla ·w_h)& \ge \xi (C_1\Vert p_h{\Vert }_0^2-C_2\Vert (\mathbb{I}-\mathsf{\Theta })p_h{\Vert }_0\Vert p_h{\Vert }_0)\hfill \\ & \ge \xi C_1\Vert p_h{\Vert }_0^2-\xi C_2\Vert (\mathbb{I}-\mathsf{\Theta })p_h{\Vert }_0{\Vert p_h\Vert }_0\hfill \\ & \ge \xi C_1\Vert p_h{\Vert }_0^2-\xi \frac{C_2^2}{{ϵ}_4}\Vert (\mathbb{I}-\mathsf{\Theta })p_h{\Vert }_0^2-\xi {ϵ}_4{\Vert p_h\Vert }_0^2.\hfill \end{array}$$

By substituting all estimates in (58), the result can be obtained.

$$\begin{array}{cc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),(-\xi {\overrightarrow{\varphi }}_h,0,0,0))\ge & -(\frac{\xi \nu {\mathsf{\Gamma }}_0^2}{{ϵ}_1}+\frac{\xi {\mathsf{\Gamma }}_0^2}{{ϵ}_2})\Vert \nabla {\overrightarrow{U}}_h{\Vert }_0^2-\frac{\xi {\mathsf{\Gamma }}_0^2}{{ϵ}_3}{\Vert curl{\overrightarrow{B}}_h\Vert }_0^2\hfill \\ & (-{ϵ}_1-{ϵ}_2-{ϵ}_3-{ϵ}_4+C_1)\xi \Vert p_h{\Vert }_0^2-\xi \frac{C_2^2}{{ϵ}_4}{\Vert (\mathbb{I}-\mathsf{\Theta })p_h\Vert }_0^2.\hfill \end{array}$$

Third term of (57).

$$\begin{array}{cc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),(0,0,-\mu {\overrightarrow{\varphi }}_{0,h},0))=& -{\nu }_m(curl{\overrightarrow{B}}_h,\mu curl{\overrightarrow{\varphi }}_{0,h})+S({\overrightarrow{U}}_h\times \overrightarrow{b},\mu {\overrightarrow{\varphi }}_{0,h})\hfill \\ & -\mu (\nabla r_h,{\overrightarrow{\varphi }}_{0,h}).\hfill \end{array}$$

(59)

It can be controlled in the following way:

$$\begin{array}{cc}\hfill \mid -({\nu }_{m}curl{\overrightarrow{B}}_h,curl\mu {\overrightarrow{\varphi }}_{0,h})\mid & \le {\nu }_m\Vert curl{\overrightarrow{B}}_h{\Vert }_0\mu {\Vert curl{\overrightarrow{\varphi }}_{0,h}\Vert }_0\hfill \\ & \le {\nu }_m\Vert curl{\overrightarrow{B}}_h{\Vert }_0\mu {\mathsf{\Gamma }}_0{\Vert r_h\Vert }_1\hfill \\ & \le \frac{{\nu }_m\mu {\mathsf{\Gamma }}_0^2}{{ϵ}_5}\Vert curl{\overrightarrow{B}}_h{\Vert }_0^2+\mu {ϵ}_5{\Vert r_h\Vert }_1^2.\hfill \end{array}$$

The Poincaré inequality and Young’s inequality are used to obtain

$$\begin{array}{cc}\hfill \mid ({\overrightarrow{U}}_h\times \overrightarrow{b},\mu {\overrightarrow{\varphi }}_h)\mid & \le \Vert {\overrightarrow{U}}_h\times \overrightarrow{b}{\Vert }_0\mu {\Vert {\overrightarrow{\varphi }}_{0,h}\Vert }_{curl}\hfill \\ & \le \mu \Vert {\overrightarrow{U}}_h{\Vert }_0{\mathsf{\Gamma }}_0{\Vert r_h\Vert }_1\hfill \\ & \le \frac{{\mathsf{\Gamma }}_0^2\mu C_P^2}{{ϵ}_6}\Vert {\overrightarrow{U}}_h{\Vert }_1^2+\mu {ϵ}_6{\Vert r_h\Vert }_1^2,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mid -\mu (\nabla r_h,{\overrightarrow{\varphi }}_{0,h})\mid & \le \xi (C_1\Vert r_h{\Vert }_0^2-C_2\Vert (\mathbb{I}-\mathsf{\Theta })r_h{\Vert }_0\Vert r_h{\Vert }_1)\hfill \\ & \le \mu C_1\Vert r_h{\Vert }_0^2-\mu C_2\Vert (\mathbb{I}-\mathsf{\Theta })r_h{\Vert }_0{\Vert r_h\Vert }_0\hfill \\ & \le \mu C_1\Vert r_h{\Vert }_0^2-\mu \frac{C_2^2}{{ϵ}_7}\Vert (\mathbb{I}-\mathsf{\Theta })r_h{\Vert }_0^2-\mu {ϵ}_7{\Vert r_h\Vert }_0^2,\hfill \end{array}$$

which concludes the third term by combining all the above inequalities in the following way:

$$\begin{array}{cc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),(0,0,-\mu {\overrightarrow{\varphi }}_{0,h},0))\ge & \frac{{\mathsf{\Gamma }}_0^2\mu C_P^2}{{ϵ}_6}\Vert {\overrightarrow{U}}_h{\Vert }_1^2-\frac{{\nu }_m\mu {\mathsf{\Gamma }}_0^2}{{ϵ}_5}{\Vert curl{\overrightarrow{B}}_h\Vert }_0^2\hfill \\ & -({ϵ}_5+{ϵ}_6-{ϵ}_7+C_1)\mu \Vert r_h{\Vert }_0^2+\mu \frac{C_2^2}{{ϵ}_7}{\Vert (I-\mathsf{\Theta })r_h\Vert }_0^2.\hfill \end{array}$$

Using Equation (57) and adding the bounded terms, we acquire:

$$\begin{array}{cc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),(& {\overrightarrow{U}}_h-\xi {\overrightarrow{\varphi }}_h,p_h,{\overrightarrow{B}}_h-\mu {\overrightarrow{\varphi }}_{0,h},r_h))\hfill \\ \hfill \ge & (\nu +\frac{{\mathsf{\Gamma }}_0^2\mu C_P^2}{{ϵ}_6}-\frac{\xi \nu {\mathsf{\Gamma }}_0^2}{{ϵ}_1}+\frac{\xi {\mathsf{\Gamma }}_0^2}{{ϵ}_2}){\Vert {\overrightarrow{U}}_h\Vert }_1^2\hfill \\ & +({\nu }_m-\frac{\xi {\mathsf{\Gamma }}_0^2}{{ϵ}_3}-\frac{{\nu }_m\mu {\mathsf{\Gamma }}_0^2}{{ϵ}_5}){\Vert curl{\overrightarrow{B}}_h\Vert }_0^2\hfill \\ & -({ϵ}_1+{ϵ}_2+{ϵ}_3+{ϵ}_4-C_1)\xi \Vert p_h{\Vert }_0^2+(1-\xi \frac{C_2^2}{{ϵ}_4}){\Vert (\mathbb{I}-\mathsf{\Theta })p_h\Vert }_0^2\hfill \\ & -({ϵ}_5+{ϵ}_6-{ϵ}_7+C_1)\mu \Vert r_h{\Vert }_0^2+(1+\mu \frac{C_2^2}{{ϵ}_7}){\Vert (\mathbb{I}-\mathsf{\Theta })r_h\Vert }_0^2\hfill \\ & \ge C_3\Vert {\overrightarrow{U}}_h{\Vert }_1^2+C_4\Vert p_h{\Vert }_0^2+C_5\Vert {\overrightarrow{B}}_h{\Vert }_0^2+C_6{\Vert r_h\Vert }_0^2\hfill \\ & \ge C^{\ast }\mid \mid \mid ({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h)\mid \mid {\mid }_{({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{S}}_h\times {\mathcal{M}}_h).}^2\hfill \end{array}$$

(60)

By using Equation (55), we obtain

$$\begin{array}{cc}\hfill \Vert \overrightarrow{U}-& \xi {\overrightarrow{\varphi }}_h{\Vert }_1+\Vert p_h{\Vert }_0+\Vert \overrightarrow{B}-\mu {\overrightarrow{\varphi }}_{0,h}{\Vert }_1+{\Vert r_h\Vert }_0\hfill \\ & \le \Vert \overrightarrow{U}{\Vert }_1+\xi \Vert {\overrightarrow{\varphi }}_h{\Vert }_0+\Vert p_h{\Vert }_0+\Vert \overrightarrow{B}{\Vert }_1+\mu \Vert {\overrightarrow{\varphi }}_{0,h}{\Vert }_0+{\Vert r_h\Vert }_0\hfill \\ & \le \Vert \overrightarrow{U}{\Vert }_1+\xi {\mathsf{\Gamma }}_0\Vert p_h{\Vert }_0+\Vert p_h{\Vert }_0+\Vert \overrightarrow{B}{\Vert }_1+\mu {\mathsf{\Gamma }}_0\Vert r_h{\Vert }_0+{\Vert r_h\Vert }_0\hfill \\ & \le C(\Vert \overrightarrow{U}{\Vert }_1+\Vert p_h{\Vert }_0+\Vert \overrightarrow{B}{\Vert }_1+\Vert r_h{\Vert }_0).\hfill \end{array}$$

(61)

Then, it would probably not be difficult to see

$$\begin{array}{cc}\hfill \mathcal{N}(({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h),& (\overrightarrow{U}-\xi {\overrightarrow{\varphi }}_h,p_h,\overrightarrow{B}-\mu {\overrightarrow{\varphi }}_{0,h},r_h))\hfill \\ & \ge {\widehat{C}}^{\ast \ast }\mid \mid \mid ({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h)\mid \mid \mid \mid \mid \mid ({\overrightarrow{U}}_h-\xi {\overrightarrow{\varphi }}_h,p_h,{\overrightarrow{B}}_h-\mu {\overrightarrow{B}}_h,r_h)\mid \mid \mid .\hfill \end{array}$$

(62)

This conclusion confirms the existence of coercivity. □

Optimal Error Estimate

In the subsequent section, the errors of the stabilized approach will be examined when applied to the MHD equation with magnetic pressure. Optimal error estimates for this modified system will be obtained.

Theorem 3. 

If $({\overrightarrow{U}}_h,p_h,{\overrightarrow{B}}_h,r_h)$ and $(\overrightarrow{U},p,\overrightarrow{B},r)$ are the solutions of Equations (27) and (17), respectively, then the following relation holds:

$$\begin{array}{c}\hfill \Vert \nabla (\overrightarrow{U}-{\overrightarrow{U}}_h){\Vert }_0+\Vert curl(\overrightarrow{B}-{\overrightarrow{B}}_h){\Vert }_0+\Vert p-p_h{\Vert }_0+{\Vert r-r_h\Vert }_0\le Ch.\end{array}$$

(63)

Proof. 

Subtracting Equation (27) from Equation (17), $\forall ({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)\in ({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{S}}_h\times {\mathcal{M}}_h),$ yields:

$$\begin{array}{ccc}\hfill \mathcal{N}\left((\overrightarrow{U}-{\overrightarrow{U}}_h,p-p_h,\overrightarrow{B}-{\overrightarrow{B}}_h,r-r_h),({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)\right)& =& G_1(p,q_h)+G_2(r,s_h).\hfill \end{array}$$

(64)

By adding and subtracting the related projection terms $({\tilde{\overrightarrow{U}}}_h,{\tilde{p}}_h,{\tilde{\overrightarrow{B}}}_h,{\tilde{r}}_h)$ and using the orthogonality, we obtain

$$\begin{array}{cc}\hfill \mathcal{N}(({\tilde{\overrightarrow{U}}}_h-{\overrightarrow{U}}_h,{\tilde{p}}_h-p_h,& {\tilde{\overrightarrow{B}}}_h-{\overrightarrow{B}}_h,{\tilde{r}}_h-r_h),({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,r_h))\hfill \\ \hfill =& \mathcal{N}(({\tilde{\overrightarrow{U}}}_h-\overrightarrow{U},{\tilde{p}}_h-p,{\tilde{\overrightarrow{B}}}_h-\overrightarrow{B},{\tilde{r}}_h-r),({\overrightarrow{u}}_h,q_h,{\overrightarrow{B}}_h,r_h))\hfill \\ & +G_1(p,q_h)+G_2(r,s_h).\hfill \end{array}$$

(65)

Utilizing (65), error orthogonality, and the weak coercivity bound (54), the following relation holds:

$$\begin{array}{cc}\hfill \mathsf{\Gamma }\mid \Vert & ({\tilde{\overrightarrow{U}}}_h-{\overrightarrow{U}}_h,{\tilde{p}}_h-p_h,{\tilde{\overrightarrow{B}}}_h-{\overrightarrow{B}}_h,{\tilde{r}}_h-r_h)\mid \Vert \hfill \\ & \le \underset{({\overrightarrow{u}}_h,q_h,{\overrightarrow{u}}_h,q_h)\in ({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{S}}_h\times {\mathcal{M}}_h)}{sup}\frac{\mathcal{N}\left(({\tilde{\overrightarrow{U}}}_h-{\overrightarrow{U}}_h,{\tilde{p}}_h-p_h,{\tilde{\overrightarrow{B}}}_h-{\overrightarrow{B}}_h,{\tilde{r}}_h-r_h),({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)\right)}{\mid \Vert ({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)\mid \Vert }\hfill \\ & =\underset{({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)\in ({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{S}}_h\times {\mathcal{M}}_h)}{sup}\frac{\mathcal{N}\left(({\tilde{\overrightarrow{U}}}_h-\overrightarrow{U},{\tilde{p}}_h-p,{\tilde{\overrightarrow{B}}}_h-\overrightarrow{B},{\tilde{r}}_h-r),({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)\right)+G_1(p,q_h)+G_2(r,s_h)}{\mid \Vert ({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,s_h)\mid \Vert }.\hfill \end{array}$$

From (56), we have that

$$\begin{array}{ccc}\hfill \mid G_1(p,q_h)+G_2(r,s_h)\mid & \le & C{G}_1{(p,p)}^{1/2}\Vert q_h{\Vert }_0+C{G}_2{(r,r)}^{1/2}{\Vert s_h\Vert }_0.\hfill \end{array}$$

(66)

From (54), we have

$$\begin{array}{cc}\hfill \mathcal{N}((& {\tilde{\overrightarrow{U}}}_h-\overrightarrow{U},{\tilde{p}}_h-p,{\tilde{\overrightarrow{B}}}_h-\overrightarrow{B},{\tilde{r}}_h-r),({\tilde{\overrightarrow{U}}}_h-{\overrightarrow{U}}_h,{\tilde{p}}_h-p_h,{\tilde{\overrightarrow{B}}}_h-{\overrightarrow{B}}_h,{\tilde{r}}_h-r_h))\hfill \\ & \le C(\Vert {\tilde{\overrightarrow{U}}}_h-\overrightarrow{U}{\Vert }_1+\Vert {\tilde{p}}_h{-p\Vert }_0+\Vert {\tilde{\overrightarrow{B}}}_h-\overrightarrow{B}{\Vert }_1+{\Vert {\tilde{r}}_h-r\Vert }_0\hfill \\ & +{\Vert (I-\mathsf{\Theta })p\Vert }_0+{\Vert (I-\mathsf{\Theta })r\Vert }_0)\mid \mid \mid ({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,r_h)\mid \mid \mid ,\hfill \end{array}$$

(67)

and consequently:

$$\begin{array}{cc}\hfill & \mathsf{\Gamma }\mid \Vert ({\tilde{\overrightarrow{U}}}_h-{\overrightarrow{U}}_h,{\tilde{p}}_h-p_h,{\tilde{\overrightarrow{B}}}_h-{\overrightarrow{B}}_h,{\tilde{r}}_h-r_h)\mid \Vert \hfill \\ & \le \underset{({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,r_h)\in ({\mathcal{X}}_h\times {\mathcal{Q}}_h\times {\mathcal{S}}_h\times {\mathcal{M}}_h)}{sup}\times \hfill \\ & \frac{C\left(\Vert {\tilde{\overrightarrow{U}}}_h-\overrightarrow{U}{\Vert }_1+\Vert {\tilde{p}}_h{-p\Vert }_0+\Vert {\tilde{\overrightarrow{B}}}_h-\overrightarrow{B}{\Vert }_1+\Vert {\tilde{r}}_h{-r\Vert }_0+\Vert (\mathbb{I}-\mathsf{\Theta })p_h{\Vert }_0+{\Vert (\mathbb{I}-\mathsf{\Theta })r_h\Vert }_0\right)\mid \mid \mid ({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,r_h)\mid \mid \mid }{\mid \mid \mid ({\overrightarrow{u}}_h,q_h,{\overrightarrow{m}}_h,r_h)\mid \mid \mid }\hfill \\ & \le \frac{C}{\mathsf{\Gamma }}(\Vert {\tilde{\overrightarrow{U}}}_h-\overrightarrow{U}{\Vert }_1+\Vert {\tilde{p}}_h{-p\Vert }_0+\Vert {\tilde{\overrightarrow{B}}}_h-\overrightarrow{B}{\Vert }_1+\Vert {\tilde{r}}_h{-r\Vert }_0+\Vert (\mathbb{I}-\mathsf{\Theta })p_h{\Vert }_0+\Vert (\mathbb{I}-\mathsf{\Theta })r_h{\Vert }_0).\hfill \end{array}$$

(68)

To complete the proof, we derive (63) by combining (68) with triangle inequality and (43)–(46). □

5. Numerical Tests

This section presents two computational tests aimed at validating the accuracy and efficiency of two stabilized inf-sup condition schemes. Specifically, we explore two established analytical solutions for the Oseen-type problem in two dimensions (2D) and assess their performance using a widely adopted benchmark problem employing MINI-type finite elements.

Utilizing the FE software Freefem++ (latest v. 4.6) [33], we conduct practical examinations of the techniques, employing UMFPACK to solve linear problems. In Example 2, we illustrate the convergence order data extracted from provided tables and utilize MATLAB (latest v. R2024a) to compare graphs depicting standard data against approximated values obtained with and without stabilization of the scheme. This example aims to provide a comprehensive understanding of the two stability conditions.

5.1. Analytical Solution Test

For numerical solutions, we consider the steady-state incompressible magnetohydrodynamic (MHD) flow through a convex domain $\mathfrak{D}={[0,1]}^2$. In order determine the numerical stability of the fully stabilized technique, we implement $p1-p1-p1-p1$ as the lowest equal order FE for all the knowns and compare the result with the standard solutions (MINI elements). A number of investigators have employed this experimental setup to solve the MHD equation [34,35], where the function $\overrightarrow{a}(x)$ represents the exact solution of velocity $\overrightarrow{U}$ and $\overrightarrow{b}(x)$ represents the magnetic field $\overrightarrow{B}$. Furthermore, the true answer for velocity $\overrightarrow{U}=(u_1,u_2)$, pressure p, magnetic field $\overrightarrow{B}=(b_1,b_2)$, and magnetic pressure r is provided as well:

$$Hydrodynamics\left\{\begin{array}{c}{u}_1=x^2{(x-1)}^{2}y(y-1)(2y-1),\hfill \\ u_2=-y^2{(y-1)}^{2}x(x-1)(2x-1),\hfill \\ p=(2x-1)(2y-1),\hfill \end{array}\right.$$

$$Magnatodynamics\left\{\begin{array}{c}{b}_1=sin(\pi x)cos(\pi y),\hfill \\ b_2=-sin(\pi y)cos(\pi x).\hfill \\ r=(2x-1)(2y-1).\hfill \end{array}\right.$$

The right sides, boundary conditions, and initial conditions are determined utilizing the model’s equations. We consider Reynolds number $R_e=1000$, Magnetic Reynolds number $R_m=1$, and the coupling term $\Psi =1$ in our experiment. However, for the concise representation, we consider $E_1=MINIElements(p1b-p1-p1-p1)$, which are standard in our experiment (

Table 1. The MHD fluid flow result’s error estimate derived from $E_1$.

h	$\left|\right|\overrightarrow{\mathit{U}}-{\overrightarrow{\mathit{U}}}_{\mathit{h}}{\left|\right|}_0$	$\left|\right|\mathit{p}-{\mathit{p}}_{\mathit{h}}{\left|\right|}_0$	$\left|\right|\overrightarrow{\mathit{B}}-{\overrightarrow{\mathit{B}}}_{\mathit{h}}{\left|\right|}_0$	$\left|\right|\mathit{r}-{\mathit{r}}_{\mathit{h}}{\left|\right|}_0$
$1/4$	0.22122	0.20562	0.12221	0.30452
$1/8$	0.10267	0.06430	0.04332	0.07430
$1/16$	0.04883	0.02187	0.01579	0.03188
$1/32$	0.02390	0.00724	0.00621	0.00734
$1/64$	0.01185	0.00247	0.00253	0.00249

), $E_2=p1-p1-p1-p1$ without the stability term (

Table 2. The MHD fluid flow result’s error estimate derived from $E_2$.

h	$\left|\right|\overrightarrow{\mathit{U}}-{\overrightarrow{\mathit{U}}}_{\mathit{h}}{\left|\right|}_0$	$\left|\right|\mathit{p}-{\mathit{p}}_{\mathit{h}}{\left|\right|}_0$	$\left|\right|\overrightarrow{\mathit{B}}-{\overrightarrow{\mathit{B}}}_{\mathit{h}}{\left|\right|}_0$	$\left|\right|\mathit{r}-{\mathit{r}}_{\mathit{h}}{\left|\right|}_0$
$1/4$	0.235998	1.78225	0.189812	1.682253
$1/8$	0.133753	0.97397	0.176594	0.454795
$1/16$	0.051102	0.56637	0.045024	0.365765
$1/32$	0.013276	0.534318	0.060156	0.214255
$1/64$	0.015074	0.257345	0.024291	0.142583

), and $E_3=p1-p1-p1-p1$ with the stability term [11] (

Table 3. The MHD fluid flow result’s error estimate derived from $E_3$.

h	$\left|\right|\overrightarrow{\mathit{U}}-{\overrightarrow{\mathit{U}}}_{\mathit{h}}{\left|\right|}_0$	$\left|\right|\mathit{p}-{\mathit{p}}_{\mathit{h}}{\left|\right|}_0$	$\left|\right|\overrightarrow{\mathit{B}}-{\overrightarrow{\mathit{B}}}_{\mathit{h}}{\left|\right|}_0$	$\left|\right|\mathit{r}-{\mathit{r}}_{\mathit{h}}{\left|\right|}_0$
$1/4$	0.235997	1.781253	0.179617	1.678122
$1/8$	0.123752	0.963972	0.066592	0.86453
$1/16$	0.061101	0.547781	0.025024	0.464631
$1/32$	0.030274	0.343097	0.010154	0.245765
$1/64$	0.014073	0.246215	0.004292	0.132873

).

5.2. Convergence Test

In Figure 2, the convergence rates for hydrodynamic $U_h$ and magnetic velocities (field) $B_h$ are demonstrated. The plots depict red lines representing the error E1 for standard MINI elements $p1b-p1-p1-p1$ in blue, E2 with stabilization term $p1-p1-p1-p1$ in green, and E3 without stabilization term $p1-p1-p1-p1$ in red. The dotted line represents the optimal error. Notably, it can be observed that error lines E1 and E2 consistently run parallel to each other, while line E3 deviates from this benchmark. From these graphs, it is clear that the scheme consistently exhibits improved stability and performance with the addition of a stabilization term in weak formulations.

In Figure 3, the convergence rates for hydrodynamic pressure $p_h$ and magnetic pressure $r_h$ are depicted. The plots represent the error lines for three different cases: E1 (standard MINI elements, $p1b-p1-p1-p1$) in red, E2 (with stabilization term, $p1-p1-p1-p1$) in blue, and E3 (without stabilization term, $p1-p1-p1-p1$) in green. The black line (doted) represents the optimal error.

The evidence for the optimal pressure error with the stabilization term is apparent in the behavior of the error lines. Particularly:

• Divergence of Non-stabilized Pressure Error: The error line E3, which represents the case without the stabilization term, diverges significantly from the other error lines. This indicates a clear lack of convergence, representing that the non-stabilized scheme fails to maintain an optimal error decay rate for pressure terms.
• Optimal Decay for Stabilized Schemes: Even for the cases with stabilization (E1 and E2), the error decay aligns with the dotted line representing the optimal error rate. Although the errors for E1 and E2 are consistently parallel to each other, suggesting similar convergence behaviors, they reach the optimal rate with the same behavior (parallel way).

Thus, the plots provide strong evidence that the stabilization term improves convergence and the stabilized schemes achieve the optimal pressure error decay for particular additional terms. This highlights the need for the stability term in the numerical approach to attain optimal convergence rates for the lowest equal order finite elements.

6. Conclusions

This work introduces a novel two-stabilized method for solving the model equations of incompressible steady magnetohydrodynamics (MHD). The method effectively presents the main question by overcoming the constraints of two inf-sup conditions through the addition of a stabilization term in the discrete variational formulation. This approach, novel for the lowest equal order finite element method, is proven to ensure the well-posedness of the scheme. The work demonstrates theoretical analysis, numerical tests, and ideal convergence rates to validate the effectiveness of the proposed scheme. The numerical examples confirm the utilization of the method in describing the complex flow characteristics in the mixed MHD model. The conclusions drawn are supported by the evidence and arguments presented, highlighting the importance of satisfying two inf-sup (or LBB) requirements for the well-posedness of the mixed MHD system in the typical Galerkin finite element method.