Multigrid computation of vector potentials

In this paper we analyze the use of edge finite elements and the multigrid method to approximate the problem of computing a static magnetic field in a cavity. We show how the mixed finite element discrete problem may be reformulated as a symmetric positive-definite system. This involves the proof of a discrete analogue of the second kind Friedrichsʹ inequality. Then we show how the multigrid method may be applied to compute the magnetic vector potential. The difficulty here is that the bilinear forms on different meshes are not related in the standard way. Two different intergrid transfer operators are defined in the paper for the multigrid method. The convergence theory and the numerical tests include both algorithms.