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

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