{"title":"Two-level Restricted Additive Schwarz preconditioner based on Multiscale Spectral Generalized FEM for Heterogeneous Helmholtz Problems","authors":"Chupeng Ma, Christian Alber, Robert Scheichl","doi":"arxiv-2409.06533","DOIUrl":null,"url":null,"abstract":"We present and analyze a two-level restricted additive Schwarz (RAS)\npreconditioner for heterogeneous Helmholtz problems, based on a multiscale\nspectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C.\nAlber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546--1584]. The\npreconditioner uses local solves with impedance boundary conditions, and a\nglobal coarse solve based on the MS-GFEM approximation space constructed from\nlocal eigenproblems. It is derived by first formulating MS-GFEM as a Richardson\niterative method, and without using an oversampling technique, reduces to the\npreconditioner recently proposed and analyzed in [Q. Hu and Z.Li, arXiv\n2402.06905]. We prove that both the Richardson iterative method and the preconditioner\nused within GMRES converge at a rate of $\\Lambda$ under some reasonable\nconditions, where $\\Lambda$ denotes the error of the underlying MS-GFEM\n\\rs{approximation}. Notably, the convergence proof of GMRES does not rely on\nthe `Elman theory'. An exponential convergence property of MS-GFEM, resulting\nfrom oversampling, ensures that only a few iterations are needed for\nconvergence with a small coarse space. Moreover, the convergence rate $\\Lambda$\nis not only independent of the fine-mesh size $h$ and the number of subdomains,\nbut decays with increasing wavenumber $k$. In particular, in the\nconstant-coefficient case, with $h\\sim k^{-1-\\gamma}$ for some $\\gamma\\in\n(0,1]$, it holds that $\\Lambda \\sim k^{-1+\\frac{\\gamma}{2}}$.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"118 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06533","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We present and analyze a two-level restricted additive Schwarz (RAS)
preconditioner for heterogeneous Helmholtz problems, based on a multiscale
spectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C.
Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546--1584]. The
preconditioner uses local solves with impedance boundary conditions, and a
global coarse solve based on the MS-GFEM approximation space constructed from
local eigenproblems. It is derived by first formulating MS-GFEM as a Richardson
iterative method, and without using an oversampling technique, reduces to the
preconditioner recently proposed and analyzed in [Q. Hu and Z.Li, arXiv
2402.06905]. We prove that both the Richardson iterative method and the preconditioner
used within GMRES converge at a rate of $\Lambda$ under some reasonable
conditions, where $\Lambda$ denotes the error of the underlying MS-GFEM
\rs{approximation}. Notably, the convergence proof of GMRES does not rely on
the `Elman theory'. An exponential convergence property of MS-GFEM, resulting
from oversampling, ensures that only a few iterations are needed for
convergence with a small coarse space. Moreover, the convergence rate $\Lambda$
is not only independent of the fine-mesh size $h$ and the number of subdomains,
but decays with increasing wavenumber $k$. In particular, in the
constant-coefficient case, with $h\sim k^{-1-\gamma}$ for some $\gamma\in
(0,1]$, it holds that $\Lambda \sim k^{-1+\frac{\gamma}{2}}$.