We derive and propose a family of new preconditioners for the saddle-point systems arising from the edge element discretization of the time-harmonic Maxwell’s equations in three dimensions. With the new preconditioners, we show that the preconditioned conjugate gradient method can apply for the saddle-point systems when wave numbers are smaller than a positive critical number, while the iterative methods like the preconditioned MINRES may apply when wave numbers are larger than the critical number. The spectral behaviors of the resulting preconditioned systems for some existing and new preconditioners are analyzed and compared, and several two-dimensional numerical experiments are presented to demonstrate and compare the efficiencies of these preconditioners.

我们推导并提出了一系列新的前置条件预处理器，用于鞍点系统，这是由时谐Maxwell方程在三个维度上的边缘元素离散化引起的。使用新的预处理器，我们显示出当波数小于正临界数时，预处理共轭梯度方法可以适用于鞍点系统，而当波数大于临界值时，像预处理MINRES这样的迭代方法可能适用数。分析和比较了一些现有的和新的预处理器的预处理系统的光谱行为，并进行了几个二维数值实验，以证明和比较这些预处理器的效率。