An Adaptive Finite Element Method for Magnetohydrodynamics

A finite element discretization for two-dimensional MHD is described. The elements are triangles with piecewise linear basis functions. The main computational difficulty is the accurate calculation of the current. The most effective solution is to employ a current–vorticity advection formulation of the equations. Acceptable results can also be obtained with a two-step calculation of the current from the vector potential. Mesh operations are described to reconnect and refine the mesh adaptively in the vicinity of nearly singular currents to improve magnetic flux conservation. Example computations of the coalescence instability, tilt mode, and divertor tokamak equilibrium, validating and illustrating the method, are presented. The simulations show the formation of current sheets, with the current density increasing exponentially in time. During this increase, the grid of initially ∼104points adapts to provide resolution comparable to a uniform grid of up to 1.6 × 108grid points.