Numerical treatment of realistic boundary conditions for the eddy current problem in an electrode via Lagrange multipliers

This paper deals with the finite element solution of the eddy current problem in a bounded conducting domain, crossed by an electric current and subject to boundary conditions appropriate from a physical point of view. Two different cases are considered depending on the boundary data: input current density flux or input current intensities. The analysis of the former is an intermediate step for the latter, which is more realistic in actual applications. Weak formulations in terms of the magnetic field are studied, the boundary conditions being imposed by means of appropriate Lagrange multipliers. The resulting mixed formulations are analyzed depending on the regularity of the boundary data. Finite element methods are introduced in each case and error estimates are proved. Finally, some numerical results to assess the effectiveness of the methods are reported.