  The H-formulation, used abundantly for the simulation of high temperature
superconductors, has shown to be a very versatile and easily implementable way
of modeling electromagnetic phenomena involving superconducting materials.
However, the simulation of a full vector field in current-free domains
unnecessarily adds degrees of freedom to the model, thereby increasing
computation times. In this contribution, we implement the well known H-$\phi$
formulation in COMSOL Multiphysics in order to compare the numerical
performance of the H and H-$\phi$ formulations in the context of computing the
magnetization of bulk superconductors. We show that the H-$\phi$ formulation
can reduce the number of degrees of freedom and computation times by nearly a
factor of two for a given relative error. The accuracy of the magnetic fields
obtained with both formulations are demonstrated to be similar. The
computational benefits of the H-$\phi$ formulation are shown to far outweigh
the added complexity of its implementation, especially in 3-D. Finally, we
identify the ideal element orders for both H and H-$\phi$ formulations to be
quartic in 2-D and cubic in 3-D, corresponding to the highest element orders
implementable in COMSOL.
