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This paper develops a weak Galerkin (WG) finite element method of arbitraryorder for the steady incompressible Magnetohydrodynamics equations. The WGscheme 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, thepressure, and the magnetic pseudo-pressure in the interior of elements, anduses piecewise polynomials of degree $k$ for their numerical traces on theinterfaces of elements. The method is shown to yield globally divergence-freeapproximations of the velocity and magnetic fields. We give existence anduniqueness results for the discrete scheme and derive optimal a priori errorestimates. We also present a convergent linearized iterative algorithm.Numerical experiments are provided to verify the obtained theoretical results.

Robust globally divergence-free Weak Galerkin finite element method for incompressible Magnetohydrodynamics flow