In this paper, we develop a steady-state magnetohydrodynamics (MHD) system coupling thermal problem with the physical parameters dependent on temperature in ${\mathbb{R}}^3$ [1,2,3], as follows: where $\mathit{n}$ is the outer unit normal of $\partial \mathsf{\Omega }$, and $\mathsf{\Omega }$ is a bounded domain with a Lipschitz boundary $\partial \mathsf{\Omega }$; ($\mathit{u}$, p, $\mathit{B}$, r, $\theta$) denote the velocity field, pressure, magnetic induction, scalar fuction and temperature; ($\nu$, $\sigma$, $\mu$, $\kappa$, $\mathit{\beta }$) denote the kinematic viscosity, electric conductivity, coupling number, thermal conductivity and thermal expansion coefficient; ($\psi$, $\mathit{f}$, $\mathit{g}$) denote a given heat source, a forcing term for magnetic induction and the known applied current with $\mathrm{div}\mathit{g}=0$. In fact, the scalar function r is a virtual function. The purpose of adding r is related to the constraint $\mathrm{div}\mathit{B}=0$ [4,5].

Many numerical methods for incompressible MHD problems have been widely studied. According to our investigation, a considerable number of scholars in the literature use continuous Lagrange finite elements to approximate velocity and magnetic induction unknowns, see, e.g., [6,7,8,9,10]. However, it is well known that using continuous elements to approximate magnetic induction may lead to an inaccurate approximation when the regularity of the magnetic unknown is lower than ${\mathit{H}}^1\left(\mathsf{\Omega }\right)$, cf. [11,12], which may often be encountered in non-convex polyhedral or with a non-$C^{1,1}$ boundary. A novel method to overcome these shortcomings was put forward in [13] by using Nédélec finite elements to approximate the magnetic unknown B. This seems attractive and has been used in [14,15,16,17] and the references therein. In addition, the authors had proposed different formulas to keep the magnetic solutions divergence-free in the numerical schemes in the papers [18,19].

Because of the movement of the fluid with viscosity, viscosity will produce heat. Under the action of the external magnetic field, the incompressible MHD problem is usually coupled with the thermal system through the famous Boussinesq approximation. For example, Meir et al. [20,21] studied the mixed finite element method for the thermally coupled MHD equation by adopting continuous elements to approximate the unknowns of magnetic, fluid and thermal problems, which is pioneering work. Coupling fluid systems or electromagnetic models with coefficients dependent on temperature face the mathematical challenge of solving strongly nonlinear partial differential equations, as studied by some physicists and mathematicians, see, e.g., [22,23,24,25,26,27,28,29] and the references therein. In addition, in many practical applications of the MHD system, the change in temperature will lead to a change in the coefficients in the fluid field and electromagnetic field. For the coupling MHD problem whose coefficients depend on temperature, it is important to study its reliable finite element method.

In this paper, we aim to give a rigorous well-posed analysis of the solution to the continuous problem and error estimates for the MHD system with variable coefficients by the finite element method. The fluid field was approximated by Taylor–Hood-type finite elements, and the thermal system was approximated by Lagrange finite elements. Considering the highly nonlinearity brought by the physical parameters dependent on temperature and the Lorentz terms in the magnetic equation, as well as a possible non-convex domain or a non-$C^{1,1}$ boundary, we choose $\mathit{H}\left(\mathbf{curl}\right)$-conforming Nédélec edge elements to approximate the magnetic equation to capture the physical solutions. The optimal error estimates of velocity, pressure, temperature, magnetic induction and Lagrange multiplier are established. As far as we know, there still lacks rigorous analysis in the literature for the error estimate of the stationary MHD thermally coupled model with temperature-dependent coefficients.

The remaining parts of this paper are organized as follows. In Section 2, we introduce some notations and basic finite element estimation used in the discussion and show the uniqueness of the solution to the continuous system. In Section 3, we propose a discrete finite element method for the variable coefficient system consisting of Equations (1)–(6). In Section 4, we give the convergence of all variables under the slightly smooth regularity assumption. In Section 5, we verify the effectiveness of the proposed numerical method through a numerical experiment.

Firstly, we introduce some symbols that will be used throughout this article. For all $m\in {\mathbb{N}}^{+}$, $1\le p\le \infty$, let $W^{m,p}\left(\mathsf{\Omega }\right)$ denote the standard Sobolev space, and when $p=2$, it can be written as $H^m\left(\mathsf{\Omega }\right)$. The notation $(·,·)$ is expressed as an inner product, namely $(\varphi ,\psi )={\int }_{\mathsf{\Omega }}\varphi \psi dx$, and the norm in $L^2\left(\mathsf{\Omega }\right)$ defined by ${\parallel ·\parallel }_0$. Vector-valued quantities will be denoted in boldface notations, such as $\mathit{u}=(u_1,u_2,u_3)$ and ${\mathit{L}}^2\left(\mathsf{\Omega }\right)={\left(L^2\left(\mathsf{\Omega }\right)\right)}^3$. We use C and c to denote generic positive constants independent of the mesh size h, and it may adopt different values in different places.

To simplify, we define the following Sobolev spaces The norm of the following types still need to be defined: We then set and $\mathcal{H}\left(\mathsf{\Omega }\right)={\mathit{W}}_0\cap \mathit{H}(\mathrm{div};\mathsf{\Omega })$, which is equipped with the following norm:

To facilitate our analysis, the following embedding results (see, e.g., Proposition 3.7 of [30] or [31]) are introduced here, which are also valid for the Lipschitz polyhedron domain.

In order to better demonstrate the stability of energy, the following trilinear terms are denoted for any $(\mathit{w},\mathit{u},\mathit{v})\in \mathit{X}\times \mathit{X}\times \mathit{X}$ and $(\theta ,\phi )\in (Y\times Y)$.

Next, the definition of the weak solution to the magneto-heat coupling system with variable coefficients (Equations (1)–(6)) is given.

In this section, we introduce a mixed finite element approximation of the MHD system coupled thermal problem in Equations (7)–(9). The approximation is based on the Nédélec first family of elements for the discretization of the magnetic induction.

Throughout, the domain $\mathsf{\Omega }$ is partitioned into a finite number of open non-overlapping subdomains with regular and quasi-uniform meshes ${\mathcal{T}}_h$ of mesh-size h that partition $\mathsf{\Omega }$ into tetrahedra K. Each tetrahedron K is supposed to be the image of a reference tetrahedron $\widehat{K}$ under an affine map $F_K$. $P_k\left(K\right)$ and ${\tilde{P}}_k\left(K\right)$ represent the space of polynomials of the total degree at most $k\ge 0$ and homogeneous polynomials k on K, respectively.

Given the generalized Taylor–Hood element $({\mathit{X}}_h^k,Q_h^{k-1})$ with $k\ge 2$, where ${\mathit{X}}_h^k$ is the k order vectorial Lagrange finite element subspace of $\mathit{X}$, and $Q_h^{k-1}$ is the $k-1$ order scalar Lagrange finite element subspace of Q. For $k=1$, the velocity and pressure pair can be approximated by the well-known stable mini-elements, cf. [31,35,36]. Furthermore, $Y_h^k$ is the k order scalar Lagrange finite element subspace of Y, refer to [31,35].

For ${\mathcal{D}}_k\left(K\right)$, it denotes the polynomials $\mathit{q}$ in ${\tilde{\mathit{P}}}_k\left(K\right)$ that satisfy $\mathit{q}\left(x\right)·\mathit{x}=0$ on K. Define the following space where $1\le k$. Using Nédélec $\mathit{H}\left(\mathbf{curl}\right)$-conforming finite element space (see [33,37]) to approximate the magnetic induction.

Setting $S_h^k=\{C\in H^1\left(\mathsf{\Omega }\right)\cap L_0^2\left(\mathsf{\Omega }\right),C\in P_k\left(K\right),\forall K\in {\mathcal{T}}_h\}$, and we can define the following weakly divergent space

In addition, the following discrete Poincaré–Friedrichs inequality is established with a constant $C_p>0$ independent of the mesh size h.

The link between the spaces ${\mathit{W}}_{0h}^k$ and $\mathcal{W}\left(\mathsf{\Omega }\right)$ is accomplished by the Hodge mapping $\mathcal{Z}:\mathit{H}(\mathbf{curl};\mathsf{\Omega })\to \mathcal{W}\left(\mathsf{\Omega }\right)$ (refer to [12]), where $\mathcal{W}\left(\mathsf{\Omega }\right)=\{\mathit{C}\in \mathcal{H}(\mathsf{\Omega }),\mathrm{div}\mathit{C}=0\mathrm{in}\mathsf{\Omega }\}$ such that Furthermore, there exists $l=l\left(\mathsf{\Omega }\right)>0$, In addition, the discrete kernel space of the divergence operator is given by

On a quasi-uniform mesh, there holds (see Theorem 3.2.6 of [38]) where $C_{inv}>0$ is a generic constant independent of the mesh size h, ı and m are two real numbers with $0\le \mathit{ı}\le m\le 1$, and p and q are two integers with $1\le p\le q\le \infty$.

The following discrete inf-sup conditions (see Chapter 2 of [36] or [12]) are founded where ${\beta }_{*}$ is a generic positive constant depending on the domain $\mathsf{\Omega }$.

For $k\ge 1$, our goal is to find $\{({\mathit{u}}_h,p_h,{\theta }_h,{\mathit{B}}_h,r_h)\in {\mathit{X}}_h^k\times Q_h^{k-1}\times Y_h^k\times {\mathit{W}}_{0h}^k\times S_h^k\}$, ∀$\{(\mathit{v},q,\phi ,\mathit{C})\in {\mathit{X}}_h^k\times Q_h^{k-1}\times Y_h^k\times {\mathit{W}}_h^k\}$ such that

In order to estimate the error in the next section, the stability of the numerical scheme in Equations (25)–(28) should be given here.

In this section, the convergence of the MHD system coupled the heat equation with variable coefficients is considered by employing the finite element method. We strictly establish optimal error estimates of velocity, pressure, temperature, magnetic induction and Lagrange multiplier under the assumption that the exact solution has low regularity.

Here, it is necessary to make the following regularity assumptions for the weak solution of Equations (7)–(9), which will facilitate the error estimate of the discrete solution.

For the convenience of the subsequent analysis, we will assume there exists a constant $C_f$ depending on $\mathit{f}$, $\mathit{g}$, $\psi$ and $\mathsf{\Omega }$ such that

Let $\ell =min\{k,s\}$, for $k\ge 1$ and $s>1/2$, with the help of [31,40], then we have the following approximation properties: where k and s are the order index of the finite element spaces and the regularity of the exact solution, respectively.

Let $(e_{\mathit{u}},e_p,e_{\theta },e_{\mathit{B}},e_r)=(\mathit{u}-{\mathit{u}}_h,p-p_h,\theta -{\theta }_h,\mathit{B}-{\mathit{B}}_h,r-r_h)$. A combination of Equations (7)–(9) and Equations (25)–(28) yields the following truncation error equations:

With the preparations of the above work, we now begin to study the optimal error estimation of each variable.