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

TL;DR

This paper introduces a high-order weak Galerkin finite element method for steady incompressible Magnetohydrodynamics that ensures divergence-free velocity and magnetic fields, with proven stability, optimal error estimates, and verified through numerical experiments.

Contribution

It develops a novel divergence-free weak Galerkin scheme for MHD equations with arbitrary order accuracy and provides theoretical analysis and practical algorithms.

Findings

The method achieves globally divergence-free velocity and magnetic fields.

Optimal a priori error estimates are established.

Numerical experiments confirm theoretical results.

Abstract

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.