A hybrid high-order scheme for the stationary, incompressible magnetohydrodynamics equations

TL;DR

This paper introduces a high-order hybrid scheme for stationary incompressible magnetohydrodynamics that is accurate on complex meshes, with proven error estimates and convergence, validated by 3D numerical tests.

Contribution

It develops a novel hybrid high-order scheme for MHD equations with rigorous error analysis and convergence proofs applicable to arbitrary polyhedral meshes.

Findings

Error estimates in energy and L2 norms are robust and optimal.

The scheme converges to the continuous solution regardless of source size.

Numerical tests confirm theoretical accuracy on tetrahedral and Voronoi meshes.

Abstract

We propose and analyse a hybrid high-order (HHO) scheme for stationary incompressible magnetohydrodynamics equations. The scheme has an arbitrary order of accuracy and is applicable on generic polyhedral meshes. For sources that are small enough, we prove error estimates in energy norm for the velocity and magnetic field, and $L^2$-norm for the pressure; these estimates are fully robust with respect to small faces, and of optimal order with respect to the mesh size. Using compactness techniques, we also prove that the scheme converges to a solution of the continuous problem, irrespective of the source being small or large. Finally, we illustrate our theoretical results through 3D numerical tests on tetrahedral and Voronoi mesh families.