  Although the H-formulation has proven to be one of the most versatile
formulations used to accurately model superconductors in the finite element
method, the use of vector dependent variables in non-conducting regions leads
to unnecessarily long computation times. Additionally, in some applications of
interest, the combination of multiple magnetic components interacting with
superconducting bulks and/or tapes leads to large domains of simulation. In
this work, we separate the magnetic field into a source and reaction field and
use the H-$\phi$ formulation to efficiently simulate a superconductor
surrounded by magnetic bodies. We model a superconducting cube between a pair
of Helmholtz coils and a permanent magnet levitating above a superconducting
pellet. In both cases, we find excellent agreement with the H-formulation,
while the computation times are reduced by factors of nearly three and four in
2-D and 3-D, respectively. Finally, we show that the H-$\phi$ formulation is
more accurate and efficient than the H-A formulation in 2-D.
