In this work we study the convergence properties of the one-level parallel Schwarz method applied to the one-dimensional and two-dimensional Helmholtz and Maxwell's equations. One-level methods are not scalable in general. However, it has recently been proven that when impedance transmission conditions are used in the case of the algorithm applied to the equations with absorption, under certain assumptions, scalability can be achieved and no coarse space is required. We show here that this result is also true for the iterative version of the method at the continuous level for strip-wise decompositions into subdomains that can typically be encountered when solving wave-guide problems. The convergence proof relies on the particular block Toeplitz structure of the global iteration matrix. Although non-Hermitian, we prove that its limiting spectrum has a near identical form to that of a Hermitian matrix of the same structure. We illustrate our results with numerical experiments.

翻译：在这项工作中,我们研究了适用于单维和二维Helmholtz和Maxwell等式的单维平行Schwarz方法的趋同特性。 单级方法一般无法伸缩。 但是,最近已经证明,在对吸收等式应用算法的情况下,如果使用阻力传输条件,根据某些假设,可以实现伸缩性,不需要粗略的空间。 我们在这里表明,对于在连续水平将条形分解成子体的方法的迭代版本,这一结果也是真实的,在解决波导问题时,通常会遇到这种迭代版本。 趋同证据依赖于全球迭代矩阵的托普利茨特定块结构。 虽然非赫米提人,但我们证明,其限制频谱与同一结构的赫米特矩阵具有几乎相同的形式。 我们用数字实验来说明我们的结果。