  We present a numerical method for the solution of linear magnetostatic
problems in domains with a symmetry direction, including axial and
translational symmetry. The approach uses a Fourier series decomposition of the
vector potential formulation along the symmetry direction and covers both,
zeroth (non-oscillatory) and non-zero (oscillatory) harmonics. For the latter
it is possible to eliminate one component of the vector potential resulting in
a fully transverse vector potential orthogonal to the transverse magnetic
field. In addition to the Poisson-like equation for the longitudinal component
of the non-oscillatory problem, a general curl-curl Helmholtz equation results
for the transverse problem covering both, non-oscillatory and oscillatory case.
The derivation is performed in the covariant formalism for curvilinear
coordinates with a tensorial permeability and symmetry restrictions on metric
and permeability tensor. The resulting variational forms are treated by the
usual nodal finite element method for the longitudinal problem and by a
two-dimensional edge element method for the transverse problem. The numerical
solution can be computed independently for each harmonic which is favourable
with regard to memory usage and parallelisation.
