Numerical analysis of a transient non-linear axisymmetric eddy current model. (October 2015) - Bermúdez, Alfredo; Gómez, Dolores; Rodríguez, Rodolfo; Venegas, Pablo
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 <alternatives><inline-graphic xlink:href="ark:/27927/pgj2svh1fx6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />B–H</alternatives> 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 <alternatives><inline-graphic xlink:href="ark:/27927/pgj2svgr1xp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />L2</alternatives> rate of convergence of order <alternatives><inline-graphic xlink:href="ark:/27927/pgj2svhbcgb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />O(h+Δt)</alternatives> without assuming any additional regularity of the solution. Moreover, under appropriate smoothness assumptions, we also prove an <alternatives><inline-graphic xlink:href="ark:/27927/pgj2svgr1xp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />L2</alternatives>-like rate of convergence of order <alternatives><inline-graphic xlink:href="ark:/27927/pgj2svh00bm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />O(h2+Δt)</alternatives>. Finally, some numerical results, which confirm the theoretically predicted behavior of the method, are reported.