On the use of the magnetic vector potential in the nodal and edge finite element analysis of 3D magnetostatic problems

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Abstract
An overview of various finite element techniques based on the magnetic vector potential for the solution of three-dimensional magnetostatic problems is presented. If nodal finite elements are used for the approximation of the vector potential, a lack of gauging results in an ill-conditioned system. The implicit enforcement of the Coulomb gauge dramatically improves the numerical stability, but the normal component of the vector potential must be allowed to be discontinuous on iron/air interfaces. If the vector potential is is interpolated with the aid of edge finite elements and no gauge is enforced, a singular system results. It can be solved efficiently by conjugate gradient methods, provided care is taken to ensure that the current density is divergence free. Finally, if a tree-cotree gauging of the vector potential is introduced, the numerical stability depends on how the tree is selected with no obvious optimal choice available.