A constrained transport divergence-free finite element method for Incompressible MHD equations

TL;DR

This paper introduces a new finite element method for 3D incompressible resistive MHD equations that ensures divergence-free solutions, improving stability and accuracy in simulating magnetohydrodynamic phenomena.

Contribution

A novel stable mixed finite element method inspired by constrained transport that guarantees divergence-free solutions for velocity, current density, and magnetic induction in MHD equations.

Findings

Proved well-posedness of the discrete solutions.

Demonstrated the quasi-optimality of the solver.

Verified theoretical results through numerical experiments.

Abstract

In this paper we study finite element method for three-dimensional incompressible resistive magnetohydrodynamic equations, in which the velocity, the current density, and the magnetic induction are divergence-free. It is desirable that the discrete solutions should also satisfy divergence-free conditions exactly especially for the momentum equations. Inspired by constrained transport method,we devise a new stable mixed finite element method that can achieve the goal. We also prove the well-posedness of the discrete solutions. To solve the resulting linear algebraic equations, we propose a GMRES solver with an augmented Lagrangian block preconditioner. By numerical experiments, we verify the theoretical results and demonstrate the quasi-optimality of the discrete solver with respect to the number of degrees of freedom