# A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D

## TL;DR

This paper introduces a robust finite element solver for 3D stationary incompressible MHD equations, featuring an efficient preconditioner that enhances convergence and scalability for practical computations.

## Contribution

It develops a new preconditioner for the finite element discretization of 3D stationary incompressible MHD equations, improving solver robustness and efficiency.

## Key findings

- Preconditioner accelerates GMRES convergence.
- Solver demonstrates robustness across numerical experiments.
- Scalability is confirmed for moderate parameters.

## Abstract

In this paper, we propose a robust solver for the finite element discrete problem of the stationary incompressible magnetohydrodynamic (MHD) equations in three dimensions. By the mixed finite element method, both the velocity and the pressure are approximated by H1-conforming finite elements, while the magnetic field is approximated by H(curl)-conforming edge elements. An efficient preconditioner is proposed to accelerate the convergence of the GMRES method for solving the linearized MHD problem. We use three numerical experiments to demonstrate the effectiveness of the finite element method and the robustness of the discrete solver. The preconditioner contains the least undetermined parameters and is optimal with respect to the number of degrees of freedom. We also show the scalability of the solver for moderate physical parameters.