## Preprint 2012-20

## Alfredo Bermúdez, Dolores Gómez, Rodolfo Rodríguez, Pilar Salgado, Pablo Venegas:

### Numerical solution of a transient non-linear axisymmetric eddy current model with non-local boundary conditions

### Abstract:

This paper deals with an axisymmetric transient eddy current problem in conductive nonlinear magnetic media. This means that the relation between the magnetic ﬁeld and the magnetic induction, the so-called H-B curve, is nonlinear. The source of the problem is the magnetic ﬂux across a meridian section of the device, which leads to a parabolic nonlinear problem with nonlocal boundary conditions. First, by applying some abstract results, we prove the existence and uniqueness of the solution to a weak formulation written in terms of the magnetic ﬁeld. Then, we compute the numerical solution of the problem by using a ﬁnite element method combined with a backward Euler time discretization. We derive error estimates in appropriate norms for both the semidiscrete (in space) and the fully discrete problems. Finally, we show numerical results which allow us to conﬁrm the theoretical estimates and to assess the performance of the proposed scheme.

This preprint gave rise to the following definitive publication(s):

**Alfredo BERMúDEZ, Dolores GóMEZ, Rodolfo RODRíGUEZ, Pilar SALGADO, Pablo VENEGAS: ***Numerical solution of a transient non-linear axisymmetric eddy current model with non-local boundary conditions*. Mathematical Models and Methods in Applied Sciences, vol. 23, 13, pp. 2495-2521, (2013).