Numerical Approximation of Maxwell Equations in Low-Frequency Regime

Abstract

This chapter deals with the Maxwell equations in low-frequency regime and their numerical approximation by finite element methods, in particular with the so called eddy current model in harmonic regime, which is standard in electrical engineering.First, a formulation in terms of the magnetic field in conductors and a scalar magnetic potential in dielectrics is considered for problems in which the current sources are boundary data. This approach allows imposing realistic (in the sense of easily measured in practice) boundary conditions. This formulation is discretized by edge elements in conductors and nodal elements in dielectrics, which leads to an important saving in computational effort. However, it needs of finding appropriate cut surfaces that make the dielectric domain simply connected, which sometimes can be very hard to implement.A couple of alternative mixed formulations that do not need such surfaces are then introduced and shown to be equivalent to the previous one. An additional advantage of these mixed formulations is that they allow dealing easily with inner current sources, too.Finally, a formulation in terms of potentials is introduced. The most expensive variable is a vector potential whose domain must contain the conductors and the support of the current source. It is proved that a nodal element discretization of this formulation converges to the correct solution if and only if this domain is chosen so that its connected components are convex polyhedra, which is not restrictive in practice.