## Preprint 2010-16

## Alfredo Bermúdez, Bibiana López-Rodríguez, Rodolfo Rodríguez, Pilar Salgado:

### Numerical solution of transient eddy current problems with input current intensities as boundary data

### Abstract:

The aim of this paper is to analyze a numerical method to solve transient eddy current problems with input current intensities as data, formulated in terms of the magnetic field in a bounded domain including conductors and dielectrics. To this end, we introduce a time-dependent weak formulation and prove its well-posedness. Under appropriate hypotheses on the input current intensities, we show that the weak solution has additional regularity and satisfies strong forms of the equations. We propose a finite element method for space discretization based on Nédélec edge elements on tetrahedral mesh, for which we prove well-posedness and error estimates. Furthermore, we introduce an implicit Euler scheme for time discretization and prove error estimates for the fully discrete problem. Moreover, a magnetic scalar potential is introduced to deal with the curl-free condition in the dielectric domain. This approach leads to an important saving in computational effort. Finally, the method is applied to solve two problems: a test with a known analytical solution and an application to electromagnetic forming.

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

**Alfredo BERMúDEZ, Bibiana LóPEZ-RODRíGUEZ, Rodolfo RODRíGUEZ, Pilar SALGADO: ***Numerical solution of transient eddy current problems with input current intensities as boundary data*. IMA Journal of Numerical Analysis, vol. 32, 3, pp. 1001-1029, (2012).