We present two new mixed finite element methods coupled with a boundary method for the three dimensional magnetostatic problem. Such formulations are obtained by coupling a finite element method inside a bounded domain with a boundary integral method involving either the Calderon equations or the inverse of Dirichlet Neumann operator to treat the exterior domain. First, we present the formulations and then prove that our mixed formulations are well posed and that they lead to a convergent Galerkin method. Finally, we give numerical results for a sphere immersed in a homogeneous (source) field in the two formulations.

SUMMARYIn this paper, we present three methods to discretize the Steklov-Poincare´operator. Two of these methods are already well known and commonly used and the third one is new. These methods are based either on the ballooning technique or on the integral theory or on the Calderon equations and we recall the principles of the discretization for each method. Then, we implement these discretization procedures in a code which treats the three-dimensional magnetostatic problem with a mixed and hybrid finite element method. The exterior domain is treated with the Steklov-Poincare´operator discretized using the three procedures. A comparison in terms of precision, performance and ease of implementation is given.