On the Stability of Mixed Finite-Element Formulations for High-Temperature Superconductors

In this article, 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.