An adaptive finite element method for magnetohydrodynamics

We describe a procedure for the adaptive h-refinement solution of the incompressible MHD equations in stream function form using a stabilized finite element formulation. The mesh is adapted based on a posteriori spatial error estimates of the magnetic field using both recovery and order extrapolation techniques. The step size for time integration is chosen so that temporal discretization errors are small relative to spatial errors. The adaptive procedure is applied to study singular current sheets in the tilt instability problem of ideal magnetohydrodynamics. Numerical results indicate a more accurate resolution of current sheets with higher-order methods than with piecewise-linear approximations.