Finite element formulation of the general magnetostatic problem in the space of solenoidal vector functions

Abstract

A new finite element method for the solution of the general magnetostatic problem is formulated and analyzed. The space of trial functions consists of solenoidal piecewise polynomial vector functions. We start with an integral formulation, in terms of the flux density, in the domain occupied by magnetic material. Using the properties [10] of the spectrum of the relevant singular integral operator we derive a weak formulation involving an integral operator on the boundary only. Thus the resulting finite element matrix consists of a sparse part corresponding to the interior of the iron domain and a full part corresponding to the boundary. Using the method of monotone operators, existence and uniqueness of the solution of the weak formulation as well as its discretization are proven. Error estimates are derived with the special emphasis on the case when magnetic permeability is large. Finally, solution of the problem by successive iteration is analyzed.