  The eddy current problem has many relevant practical applications in science,
ranging from non-destructive testing to magnetic confinement of plasma in
fusion reactors. It arises when electrical conductors are immersed in an
external time-varying magnetic field operating at frequencies for which
electromagnetic wave propagation effects can be neglected.
  Popular formulations of the eddy current problem either use the magnetic
vector potential or the magnetic scalar potential. They have individual
advantages and disadvantages. One challenge is related to differential
geometry: Scalar potential based formulations run into trouble when conductors
are present in non-trivial topology, as approximation spaces must be then
augmented with generators of the first cohomology group of the non-conducting
domain.
  For all existing algorithms based on lowest order methods it is assumed that
the extension of the graph-based algorithms to high-order approximations
requires hierarchical bases for the curl-conforming discrete spaces. However,
building on insight on de Rham complexes approximation with splines, we will
show in the present submission that the hierarchical basis condition is not
necessary. Algorithms based on spanning tree techniques can instead be adapted
to work on an underlying hexahedral mesh arising from isomorphisms between
spline spaces of differential forms and de Rham complexes on an auxiliary
control mesh.
