Numerical analysis of a transient non-linear axisymmetric eddy current model

Abstract

This paper deals with the numerical solution of an axisymmetric transient eddy current problem in a conductive non-linear magnetic media. This means that the relation between the magnetic field and the magnetic induction (i.e., the so-called B–H curve) is non-linear. We analyze a weak formulation of the resulting problem in the axisymmetric case, with the source term given by means of a non-homogeneous Dirichlet boundary condition. For its numerical approximation, we propose a fully discrete scheme based on a finite element method combined with a backward Euler time discretization. We establish its well-posedness and derive error estimates in appropriate norms for the proposed scheme. In particular, we obtain an L2 rate of convergence of order O(h+Δt) without assuming any additional regularity of the solution. Moreover, under appropriate smoothness assumptions, we also prove an L2-like rate of convergence of order O(h2+Δt). Finally, some numerical results, which confirm the theoretically predicted behavior of the method, are reported.