Existence and approximation of a mixed formulation for thin film magnetization problems in superconductivity

Abstract

We recall a recently introduced mixed formulation of thin film magnetization problems for type-II superconductors written in terms of two variables, the electric field and the magnetization function, see [Electric field formulation for thin film magnetization problems, Supercond. Sci. Technol.25 (2012) 104002]. A finite element approximation, [Formula: see text], based on this mixed formulation, involving the lowest-order Raviart–Thomas element for approximating the electric field, was also introduced in [Electric field formulation for thin film magnetization problems, Supercond. Sci. Technol.25 (2012) 104002]. Here h, τ are the spatial and temporal discretization parameters, and [Formula: see text] with p-1 the value of power in the current–voltage relation characterizing the superconducting material. In this paper, we establish well-posedness of [Formula: see text], and prove convergence of the unique solution of [Formula: see text] to a solution of the power law model ( Q r), for a fixed r > 1, as h, τ → 0. In addition, we prove convergence of a solution of ( Q r) to a solution of the critical state model (Q), as r → 1. Hence, we prove existence of solutions to ( Q r), for a fixed r > 1, and (Q). Finally, numerical experiments are presented.