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The eddy current problem has many relevant practical applications in science,ranging from non-destructive testing to magnetic confinement of plasma infusion reactors. It arises when electrical conductors are immersed in anexternal time-varying magnetic field operating at frequencies for whichelectromagnetic wave propagation effects can be neglected.  Popular formulations of the eddy current problem either use the magneticvector potential or the magnetic scalar potential. They have individualadvantages and disadvantages. One challenge is related to differentialgeometry: Scalar potential based formulations run into trouble when conductorsare present in non-trivial topology, as approximation spaces must be thenaugmented with generators of the first cohomology group of the non-conductingdomain.  For all existing algorithms based on lowest order methods it is assumed thatthe extension of the graph-based algorithms to high-order approximationsrequires hierarchical bases for the curl-conforming discrete spaces. However,building on insight on de Rham complexes approximation with splines, we willshow in the present submission that the hierarchical basis condition is notnecessary. Algorithms based on spanning tree techniques can instead be adaptedto work on an underlying hexahedral mesh arising from isomorphisms betweenspline spaces of differential forms and de Rham complexes on an auxiliarycontrol mesh.

Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems