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.

翻译：非线性静磁学问题的数值求解通常基于磁势的变分公式、有限元离散化以及牛顿法等迭代求解器。矢量势方法旨在最小化特定能量泛函，在三维问题中需要使用棱边单元与适当的规范条件。而标量势方法则寻求最大化负余能，可通过标准拉格朗日有限元实现，从而减少自由度数量并简化实现。然而，已有研究表明求解非线性控制系统所需的牛顿迭代次数通常高于矢量势公式。本文提出一种融合两者优势的方法，即所需牛顿迭代次数与矢量势公式相当，同时以磁标量势作为主要未知量。我们讨论了该方法的变分基础、适定性及高效实现。通过数值算例展示该方法的精度及相较其他方法的效率提升。