  In this work, we present a parallel, fully-distributed finite element
numerical framework to simulate the low-frequency electromagnetic response of
superconducting devices, which allows to efficiently exploit HPC platforms. We
select the so-called H-formulation, which uses the magnetic field as a state
variable. N\'ed\'elec elements (of arbitrary order) are required for an
accurate approximation of the H-formulation for modelling electromagnetic
fields along interfaces between regions with high contrast medium properties.
An h-adaptive mesh refinement technique customized for N\'ed\'elec elements
leads to a structured fine mesh in areas of interest whereas a smart coarsening
is obtained in other regions. The composition of a tailored, robust, parallel
nonlinear solver completes the exposition of the developed tools to tackle the
problem. First, a comparison against experimental data is performed to show the
availability of the finite element approximation to model the physical
phenomena. Then, a selected state-of-the-art 3D benchmark is reproduced,
focusing on the parallel performance of the algorithms.
