On the solvability of magnetostatic vector potential formulations

A number of existing numerical vector potential formulations are examined using a general framework described in terms of an error-based approach to computational electromagnetics together with a theorem on the uniqueness of vector distributions. It is shown that a unique vector potential is determined not only by the specification of its curl and divergence but also by proper specification of continuity and boundary conditions. The latter are only partially determined by the physical specifications of the problem, so that additional conditions must be introduced to secure solvability. The available choices are discussed, and it is shown that, in certain cases, care must be exercised to avoid inadvertent violation of physical specifications and to avoid overconstraining the trail functions unduly. It is also shown that appending the gauge error to the physically based constitutive error yields a general framework that encompasses a number of existing treatments. The framework provides a tangible basis for assessment and comparison