Volume 55, pp. 112-141, 2022.

Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices

Niall Bootland, Victorita Dolean, Alexander Kyriakis, and Jennifer Pestana


In this work we study the convergence properties of the one-level parallel Schwarz method with Robin transmission conditions 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 being applied to the equations with absorption, then, 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 are typically 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.

Full Text (PDF) [499 KB], BibTeX

Key words

domain decomposition methods, Helmholtz equations, Maxwell equations, Schwarz algorithms, one-level methods, block Toeplitz matrices

AMS subject classifications

65N55, 65N35, 65F10, 15A18, 15B05

Links to the cited ETNA articles

[9]Vol. 49 (2018), pp. 210-243 Gabriele Ciaramella and Martin J. Gander: Analysis of the parallel Schwarz method for growing chains of fixed-sized subdomains: Part III

< Back