The magnetostatic field distribution in a nonlinear medium amounts to the unique minimizer of the magnetic coenergy over all fields that can be generated by the same current. This is a nonlinear saddlepoint problem whose numerical solution can in principle be achieved by mixed finite element methods and appropriate nonlinear solvers. The saddlepoint structure, however, makes the solution cumbersome. A remedy is to split the magnetic field into a known source field and the gradient of a scalar potential which is governed by a convex minimization problem. The penalty approach avoids the use of artificial potentials and Lagrange multipliers and leads to an unconstrained convex minimization problem involving a large parameter. We provide a rigorous justification of the penalty approach by deriving error estimates for the approximation due to penalization. We further highlight the close connections to the Lagrange-multiplier and scalar potential approach. The theoretical results are illustrated by numerical tests for a typical benchmark problem

翻译：非线性介质中的静磁场分布等价于在所有由同一电流产生的磁场中，磁共能唯一极小化的问题。这是一个非线性鞍点问题，原则上可通过混合有限元方法和适当的非线性求解器实现数值求解。然而，鞍点结构使得求解过程繁琐。一种解决方案是将磁场分解为已知源场和标量势的梯度，其中标量势由凸极小化问题控制。罚函数方法避免了使用人工势和拉格朗日乘子，并导出一个涉及大参数的无约束凸极小化问题。我们通过推导因罚函数近似产生的误差估计，为罚函数方法提供了严格的理论依据。此外，我们进一步阐明了该方法与拉格朗日乘子法和标量势方法之间的紧密联系。理论结果通过典型基准问题的数值测试进行了验证。