A Quasi-variational Inequality Problem in Superconductivity

We derive a class of analytical solutions and a dual formulation of a scalar two-space-dimensional quasi-variational inequality problem in applied superconductivity. We approximate this formulation by a fully practical ̄nite element method based on the lowest order Raviart Thomas element, which yields approximations to both the primal and dual variables (the magnetic and electric ̄elds). We prove the subsequence convergence of this approximation, and hence prove the existence of a solution to both the dual and primal formulations, for strictly star-shaped domains. The e®ectiveness of the approximation is illustrated by numerical examples with and without this domain restriction.