Analysis of an ungauged T, $\phi$–$\phi$ formulation of the eddy current problem with currents and voltage excitations

Bermúdez, Alfredo; Piñeiro, Marta; Rodríguez, Rodolfo; Salgado, Pilar

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2487-2509.

The objective of this work is the analysis of a time-harmonic eddy current problem with prescribed currents or voltage drops on the boundary of the conducting domain. We will focus on an ungauged formulation that splits the magnetic field into three terms: a vector potential $𝐓$, defined in the conducting domain, a scalar potential $\phi$, supported in the whole domain, and a linear combination of source fields, only depending on the geometry. To compute the source field functions we make use of the analytical expression of the Biot−Savart law in the dielectric domain. The most important advantage of this methodology is that it eliminates the need of multivalued scalar potentials. Concerning the discretisation, edge finite elements will be employed for the approximation of both the source field and the vector potential, and standard Lagrange finite elements for the scalar potential. To perform the analysis, we will establish an equivalence between the $𝐓$,$\phi$–$\phi$ formulation of the problem and a slight variation of a magnetic field formulation whose well-possedness has already been proved. This equivalence will also be the key to prove convergence results for the discrete scheme. Finally, we will present some numerical results that corroborate the analytical ones.

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