In this paper we mathematically analyse an evolution variational
inequality which formulates the double critical-state model for type-II
superconductivity in 3D space and propose a finite element method to
discretize the formulation. The double critical-state model
originally proposed by Clem and Perez-Gonzalez is
formulated as a model in 3D space which characterizes the nonlinear
relation between the electric field, the electric current, the
perpendicular component of the electric current to the magnetic flux,
and the parallel component of the current to the magnetic flux in bulk type-II superconductor. The existence of a solution to
the variational inequality formulation is proved and the representation
theorem of subdifferential for a class of energy functionals including our
energy is established. The variational inequality formulation is
discretized in time by a semi-implicit scheme and in
space by the edge finite element of lowest order on a tetrahedral mesh.
The fully discrete formulation is an unconstrained optimisation problem.
The subsequence convergence property of the fully discrete solution is
proved. Some numerical results computed under a rotating applied magnetic
field are presented.