A reduced scalar potential approach for magnetostatics avoiding the coenergy

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.