Robust globally divergence-free HDG finite element method for steady thermally coupled incompressible MHD flow

TL;DR

This paper introduces a high-order hybridizable discontinuous Galerkin method for steady thermally coupled incompressible MHD flow that ensures divergence-free velocity and magnetic fields, with proven stability and accuracy.

Contribution

It develops a novel HDG finite element scheme of arbitrary order that guarantees divergence-free solutions for MHD flows, with rigorous theoretical analysis and numerical validation.

Findings

The method achieves optimal error estimates.

It produces globally divergence-free velocity and magnetic fields.

Numerical experiments confirm theoretical predictions.

Abstract

This paper develops an hybridizable discontinuous Galerkin (HDG) finite element method of arbitrary order for the steady thermally coupled incompressible Magnetohydrodynamics (MHD) flow. The HDG scheme uses piecewise polynomials of degrees $k(k\geq 1),k,k-1,k-1$, and $k$ respectively for the approximations of the velocity, the magnetic field, the pressure, the magnetic pseudo-pressure, and the temperature 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. Existence and uniqueness results for the discrete scheme are given and optimal a priori error estimates are derived. Numerical experiments are provided to verify the obtained theoretical results.