Abstract Time-harmonic problems arise in many important applications, such as eddy current optimally controlled electromagnetic problems. Eddy current modelling can also be used in non-destructive testings of conducting materials. Using a truncated Fourier series to approximate the solution, for linear problems the equation for different frequencies separate, so it suffices to study solution methods for the problem for a single frequency. The arising discretized system takes a two-by-two or four-by-four block matrix form. Since the problems are in general three-dimensional in space and hence of very large scale, one must use an iterative solution method. It is then crucial to construct efficient preconditioners. It is shown that an earlier used preconditioner for optimal control problems is applicable here also and leads to very tight eigenvalue bounds and hence very fast convergence such as for a Krylov subspace iterative solution method. A comparison is done with an earlier used block diagonal preconditioner.

中文翻译：

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涡流最优控制时谐电磁问题的预处理方法

摘要 时谐问题出现在许多重要的应用中，例如涡流优化控制的电磁问题。涡流建模还可用于导电材料的无损检测。用截断傅立叶级数逼近解，对于线性问题，不同频率的方程是分开的，所以研究单频率问题的求解方法就足够了。出现的离散化系统采用 2×2 或 4×4 块矩阵形式。由于问题通常在空间上是三维的，因此规模非常大，因此必须使用迭代求解方法。因此，构建高效的预处理器至关重要。结果表明，早期用于最优控制问题的预处理器也适用于这里，并导致非常严格的特征值界限，因此收敛速度非常快，例如 Krylov 子空间迭代求解方法。与较早使用的块对角预处理器进行比较。