We consider the convergence of iterative solvers for problems of nonlinear magnetostatics. Using the equivalence to an underlying minimization problem, we can establish global linear convergence of a large class of methods, including the damped Newton-method, fixed-point iteration, and the Kacanov iteration, which can all be interpreted as generalized gradient descent methods. Armijo backtracking isconsidered for an adaptive choice of the stepsize. The general assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, as well as permanent magnets. The main results are proven on the continuous level, but they carry over almost verbatim to various approximation schemes, including finite elements and isogeometric analysis, leading to bounds on the iteration numbers, which are independent of the particular discretization. The theoretical results are illustrated by numerical tests for a typical benchmark problem.

翻译：我们考虑非线性静磁问题迭代求解器的收敛性。利用与底层极小化问题的等价性，我们可建立一大类方法的全局线性收敛性，包括阻尼牛顿法、不动点迭代和Kacanov迭代，它们均可解释为广义梯度下降法。采用Armijo回溯法自适应选择步长。本分析所需的一般假设涵盖了非均匀、非线性和各向异性材料以及永磁体。主要结果在连续层面上得到证明，但几乎逐字适用于各种近似方案，包括有限元和等几何分析，从而得到与特定离散化无关的迭代次数界限。通过典型基准问题的数值测试说明了理论结果。