## Preprint 2013-09

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

### Mathematical and numerical analysis of a transient non-linear axisymmetric eddy current model

### Abstract:

This paper deals with the analysis and computation of transient electromagnetic fields in conductive non-linear magnetic media. 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. Existence and uniqueness of solution is proved for this problem under fairly general assumptions on the non-linear constitutive relation between the magnetic field H and the magnetic induction B. The technique we use is based on implicit time discretization, a priori estimates and passage to the limit by compactness. For the numerical approximation, we propose a backward Euler time-discretization and prove its well-posedness and error estimates. Subsequently, we combine it with a finite element method for space discretization. Stability and error estimates are also obtained for this full-discretization. Finally, some numerical results, which confirm the theoretically predicted behavior of the method, are reported.

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

**Alfredo BERMúDEZ, Dolores GóMEZ, Rodolfo RODRíGUEZ, Pablo VENEGAS: ***Numerical analysis of a transient non-linear axisymmetric eddy current model*. Computers & Mathematics with Applications, vol. 70, 8, pp. 1984-2005, (2015).