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SUMMARYIn field systems containing a divergenceless vector, the problem may be posed in terms of a vector potential for convenience. For the solution of the magnetic vector potential in three dimensional problems with current sources, there exist three standard variational formulations in the literature. While all these are known to give verifiable physical solutions, there is some question as to which is to be preferred. Indeed, one of them is invalid for infinite dimensional fields in that, without the finite element trial functions, it will not give unique solutions since it does not explicitly impose the divergence of the vector potential.In this paper, we look at the formulations in the light of the restrictions imposed by the finite element trial functions for tetrahedral elements and arrive at the curious result that that formulation which is totally invalid when the vector potential is unrestricted by trial functions, is in fact valid in finite element analysis and, at the same time, is the best. We further show that this formulation yields naturally non-divergent vector potential solutions, strictly as a result of the trial functions.

Adaptive refinement of finite element meshes has been performed using the bending of electromagnetic fields at element edges as a measure of error, in problems wih translational symmetry. In this paper it is presented how this could be extended to axisymmetric field problems. It is shown that the error estimate given by the inverse tan relationship is the same in electric and magnetic field problems, although in the governing Poisson equation, permittivity appears in the numerator and permeability in the denominator. This allows the same program developed for magnetic field problems to be used with minor modifications for axisymmetric electric field problems.

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