If a superconductor is cooled down below a certain critical temperature, it loses its electrical resistivity. Based on this property, a superconductor can transfer an electric current without dissipation. Moreover, the second underlying property of superconductivity is the Meissner-Ochsenfeld effect: If a weak external magnetic field is applied to a superconductor below its critical temperature, the magnetic flux is completely expelled from the material. Nowadays, many modern technological applications rely on superconductors including Magnetic Resonance Imaging (MRI), magnetic confinement fusion technologies and magnetic levitation technologies. Thus, theoretical understanding of this topic is of high academic interest. From a mathematical point of view, this phenomenon is described by Maxwell’s equations combined with a set of additional assumptions, which make the mathematical analysis very challenging and require deep knowledge in the field of computational electromagnetism. As a result, our research revolves around hyperbolic (quasi-)variational inequalities governed by the evolutionary Maxwell’s equations. We use regularization techniques as well as finite element methods to present a rigorous analysis and applicable numerical results.