Preconditioners for Mixed Finite Element Discretizations of Incompressible MHD Equations

We consider preconditioning techniques for a mixed finite element discretization of an incompressible magnetohydrodynamics (MHD) problem. Upon discretization and linearization, a 4-by-4 nonsymmetric block-structured linear system needs to be (repeatedly) solved. One of the principal challenges is the presence of a skew-symmetric term that couples the fluid velocity with the magnetic field. We propose a preconditioner that exploits the block structure of the underlying linear system, utilizing and combining effective solvers for the mixed Maxwell and the Navier--Stokes subproblems. We perform a spectral analysis for an ideal version of the preconditioner, and develop and test a practical version of it. Large-scale numerical results for linear systems of dimensions up to $10^7$ in two and three dimensions validate the effectiveness of our approach.