  In this work, we present and analyze the numerical stability of two coupled
finite element formulations. The first one is the h-a-formulation and is well
suited for modeling systems with superconductors and ferromagnetic materials.
The second one, the so-called t-a-formulation with thin-shell approximation,
applies for systems with thin superconducting domains. Both formulations
involve two coupled unknown fields and are mixed on the coupling interfaces.
Function spaces in mixed formulations must satisfy compatibility conditions to
ensure stability of the problem and reliability of the numerical solution. We
propose stable choices of function spaces using hierarchical basis functions
and demonstrate the effectiveness of the approach on simple 2D examples.
