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

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Abstract

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.
Original languageEnglish
Pages (from-to)112-141
Number of pages30
JournalETNA - Electronic Transactions on Numerical Analysis
Volume55
DOIs
Publication statusPublished - 10 Nov 2021

Keywords

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

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