This paper develops a weak Galerkin (WG) finite element method of arbitrary order for the steady incompressible Magnetohydrodynamics equations. The WG scheme uses piecewise polynomials of degrees $k(k\geq 1),k,k-1$, and $k-1$ respectively for the approximations of the velocity, the magnetic field, the pressure, and the magnetic pseudo-pressure in the interior of elements, and uses piecewise polynomials of degree $k$ for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. We give existence and uniqueness results for the discrete scheme and derive optimal a priori error estimates. We also present a convergent linearized iterative algorithm. Numerical experiments are provided to verify the obtained theoretical results.

翻译：本文针对稳态不可压缩磁流体动力学方程组，发展了一种任意阶的弱伽辽金有限元方法。该WG格式在单元内部采用次数分别为$k(k\geq 1)、k、k-1$和$k-1$的分片多项式来近似速度、磁场、压力和磁拟压力，在单元界面上则采用次数为$k$的分片多项式表示其数值迹。理论证明该方法能得到速度场和磁场全局散度自由的近似解。我们给出了离散格式的存在唯一性结果，并推导了最优先验误差估计。同时提出了一种收敛的线性化迭代算法，最后通过数值实验验证了所获得的理论结果。