  The numerical solution of problems in nonlinear magnetostatics is typically
based on a variational formulation in terms of magnetic potentials, the
discretization by finite elements, and iterative solvers like the Newton
method. The vector potential approach aims at minimizing a certain energy
functional and, in three dimensions, requires the use of edge elements and
appropriate gauging conditions. The scalar potential approach, on the other
hand, seeks to maximize the negative coenergy and can be realized by standard
Lagrange finite elements, thus reducing the number of degrees of freedom and
simplifying the implementation. The number of Newton iterations required to
solve the governing nonlinear system, however, has been observed to be usually
higher than for the vector potential formulation. In this paper, we propose a
method that combines the advantages of both approaches, i.e., it requires as
few Newton iterations as the vector potential formulation while involving the
magnetic scalar potential as the primary unknown. We discuss the variational
background of the method, its well-posedness, and its efficient implementation.
Numerical examples are presented for illustration of the accuracy and the gain
in efficiency compared to other approaches.
