We propose a finite element method for the three-dimensional transient incompressible magnetohydrodynamic equations that ensures exactly divergence-free approximations of the velocity and the magnetic induction. We employ second-order semi-implicit timestepping, for which we rigorously establish an energy law and, as a consequence, unconditional stability. We prove unique solvability of the linear systems of equations to be solved in every timestep. For those we design an efficient preconditioner so that the number of preconditioned GMRES iterations is uniformly bounded with respect to the number of degrees of freedom. As both meshwidth and timestep size tend to zero, we prove that the discrete solutions converge to a weak solution of the continuous problem. Finally, by several numerical experiments, we confirm the predictions of the theory and demonstrate the efficiency of the preconditioner.

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 H 1 (Ω)-conforming finite elements, while the magnetic field is approximated by H(curl, Ω)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.