Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices
Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices
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DOI：
10.1553/etna_vol55s112
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发表时间：
2020-06
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影响因子：
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通讯作者：
N. Bootland;V. Dolean;Alexandros Kyriakis;J. Pestana
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文献类型：
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作者：
N. Bootland;V. Dolean;Alexandros Kyriakis;J. Pestana
本文研究了应用于一维和二维亥姆霍兹方程和麦克斯韦方程的一能级平行Schwarz方法的收敛性。一级方法通常是不可伸缩的。然而，最近已经证明，当该算法应用于具有吸收的方程时，在阻抗传输条件下，在一定的假设下，可以实现可扩展性，并且不需要粗糙空间。我们在这里表明，这一结果也适用于该方法的迭代版本在连续水平上的条带分解到子域，这在解决波导问题时通常会遇到。收敛性证明依赖于全局迭代矩阵的特定块Toeplitz结构。虽然它是非厄米矩阵，但我们证明了它的极限谱与具有相同结构的厄米矩阵的极限谱具有近似相同的形式。我们用数值实验来说明我们的结果。
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.