{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T14:17:43Z","timestamp":1776867463305,"version":"3.51.2"},"reference-count":38,"publisher":"American Mathematical Society (AMS)","issue":"339","license":[{"start":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T00:00:00Z","timestamp":1696550400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/S003975\/1"],"award-info":[{"award-number":["EP\/S003975\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/R005591\/1"],"award-info":[{"award-number":["EP\/R005591\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/R005591\/1"],"award-info":[{"award-number":["EP\/R005591\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/S003975\/1"],"award-info":[{"award-number":["EP\/S003975\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/S003975\/1"],"award-info":[{"award-number":["EP\/S003975\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/R005591\/1"],"award-info":[{"award-number":["EP\/R005591\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/R005591\/1"],"award-info":[{"award-number":["EP\/R005591\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/S003975\/1"],"award-info":[{"award-number":["EP\/S003975\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    The Restricted Additive Schwarz method with impedance transmission conditions, also known as the Optimised Restricted Additive Schwarz (ORAS) method, is a simple overlapping one-level parallel domain decomposition method, which has been successfully used as an iterative solver and as a preconditioner for discretised Helmholtz boundary-value problems. In this paper, we give, for the first time, a convergence analysis for ORAS as an iterative solver\u2014and also as a preconditioner\u2014for nodal finite element Helmholtz systems of any polynomial order. The analysis starts by showing (for general domain decompositions) that ORAS is an unconventional finite element approximation of a classical parallel iterative Schwarz method, formulated at the PDE (non-discrete) level. This non-discrete Schwarz method was recently analysed in [Gong, Gander, Graham, Lafontaine, and Spence,\n                    <italic>Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation<\/italic>\n                    ], and the present paper gives a corresponding discrete version of this analysis. In particular, for domain decompositions in strips in 2-d, we show that, when the mesh size is small enough, ORAS inherits the convergence properties of the Schwarz method, independent of polynomial order. The proof relies on characterising the ORAS iteration in terms of discrete \u2018impedance-to-impedance maps\u2019, which we prove (via a novel weighted finite-element error analysis) converge as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h right-arrow 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">h\\rightarrow 0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the operator norm to their non-discrete counterparts.\n                  <\/p>","DOI":"10.1090\/mcom\/3772","type":"journal-article","created":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T12:14:00Z","timestamp":1665058440000},"page":"175-215","source":"Crossref","is-referenced-by-count":11,"title":["Convergence of restricted additive Schwarz with impedance transmission conditions for discretised Helmholtz problems"],"prefix":"10.1090","volume":"92","author":[{"given":"Shihua","family":"Gong","sequence":"first","affiliation":[]},{"given":"Ivan","family":"Graham","sequence":"additional","affiliation":[]},{"given":"Euan","family":"Spence","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,10,6]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1006\/jcph.1997.5742","article-title":"A domain decomposition method for the Helmholtz equation and related optimal control problems","volume":"136","author":"Benamou, Jean-David","year":"1997","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"320","key":"2","doi-asserted-by":"publisher","first-page":"2559","DOI":"10.1090\/mcom\/3447","article-title":"Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption","volume":"88","author":"Bonazzoli, M.","year":"2019","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/j.camwa.2021.07.011","article-title":"A comparison of coarse spaces for Helmholtz problems in the high frequency regime","volume":"98","author":"Bootland, Niall","year":"2021","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"issue":"5","key":"4","doi-asserted-by":"publisher","first-page":"1803","DOI":"10.1051\/m2an\/2018031","article-title":"High-frequency behaviour of corner singularities in Helmholtz problems","volume":"52","author":"Chaumont-Frelet, T.","year":"2018","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/2822-7840","issn-type":"print"},{"issue":"5","key":"5","doi-asserted-by":"publisher","first-page":"2445","DOI":"10.3934\/cpaa.2020107","article-title":"Uniform a priori estimates for elliptic problems with impedance boundary conditions","volume":"19","author":"Chaumont-Frelet, Th\u00e9ophile","year":"2020","journal-title":"Commun. Pure Appl. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1534-0392","issn-type":"print"},{"issue":"4","key":"6","doi-asserted-by":"publisher","first-page":"2331","DOI":"10.1137\/130917144","article-title":"A source transfer domain decomposition method for Helmholtz equations in unbounded domain","volume":"51","author":"Chen, Zhiming","year":"2013","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1007\/s00211-022-01288-x","article-title":"Robust treatment of cross-points in optimized Schwarz methods","volume":"151","author":"Claeys, Xavier","year":"2022","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"8","isbn-type":"print","volume-title":"M\\'{e}thodes de d\\'{e}composition de domaine pour les probl\\`emes de propagation d'ondes en r\\'{e}gime harmonique. Le th\\'{e}or\\`eme de Borg pour l'\\'{e}quation de Hill vectorielle","author":"Despr\u00e9s, Bruno","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/2726107060"},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"779","DOI":"10.1007\/s00211-021-01251-2","article-title":"Corners and stable optimized domain decomposition methods for the Helmholtz problem","volume":"149","author":"Despr\u00e9s, B.","year":"2021","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"10","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974065.ch1","volume-title":"An introduction to domain decomposition methods","author":"Dolean, Victorita","year":"2015","ISBN":"https:\/\/id.crossref.org\/isbn\/9781611974058"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"782","DOI":"10.1137\/140953125","article-title":"Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number","volume":"53","author":"Du, Yu","year":"2015","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"150","key":"12","doi-asserted-by":"publisher","first-page":"441","DOI":"10.2307\/2006095","article-title":"Polynomial approximation of functions in Sobolev spaces","volume":"34","author":"Dupont, Todd","year":"1980","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"13","doi-asserted-by":"publisher","first-page":"945","DOI":"10.1023\/B:BITN.0000014563.33622.1d","article-title":"Why restricted additive Schwarz converges faster than additive Schwarz","volume":"43","author":"Efstathiou, Evridiki","year":"2003","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"issue":"2","key":"14","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1137\/100804644","article-title":"Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers","volume":"9","author":"Engquist, Bj\u00f6rn","year":"2011","journal-title":"Multiscale Model. Simul.","ISSN":"https:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"issue":"4","key":"15","doi-asserted-by":"publisher","first-page":"1471","DOI":"10.1137\/040615195","article-title":"A novel multigrid based preconditioner for heterogeneous Helmholtz problems","volume":"27","author":"Erlangga, Y. A.","year":"2006","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"4","key":"16","doi-asserted-by":"publisher","first-page":"2872","DOI":"10.1137\/080737538","article-title":"Discontinuous Galerkin methods for the Helmholtz equation with large wave number","volume":"47","author":"Feng, Xiaobing","year":"2009","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"276","key":"17","doi-asserted-by":"publisher","first-page":"1997","DOI":"10.1090\/S0025-5718-2011-02475-0","article-title":"\u210e\ud835\udc5d-discontinuous Galerkin methods for the Helmholtz equation with large wave number","volume":"80","author":"Feng, Xiaobing","year":"2011","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"18","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1007\/s00211-015-0700-2","article-title":"Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?","volume":"131","author":"Gander, M. J.","year":"2015","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1137\/16M109781X","article-title":"A class of iterative solvers for the Helmholtz equation: factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods","volume":"61","author":"Gander, Martin J.","year":"2019","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"key":"20","doi-asserted-by":"crossref","unstructured":"S. Gong, M. J. Gander, I. G. Graham, D. Lafontaine, and E. A. Spence, Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation, Numer. Math., to appear,  arXiv:2106.05218, 2021.","DOI":"10.1007\/s00211-022-01318-8"},{"key":"21","doi-asserted-by":"crossref","unstructured":"S. Gong, M. J. Gander, I. G. Graham, and E. A. Spence, A variational interpretation of restricted additive Schwarz with impedance transmission condition for the Helmholtz problem, Proceedings of 26th Domain Decomposition Conference, 279\u2013286  arXiv:2103.11379, 2022.","DOI":"10.1007\/978-3-030-95025-5_30"},{"issue":"3","key":"22","doi-asserted-by":"publisher","first-page":"2139","DOI":"10.1093\/imanum\/draa080","article-title":"Domain decomposition preconditioners for high-order discretizations of the heterogeneous Helmholtz equation","volume":"41","author":"Gong, Shihua","year":"2021","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"5","key":"23","doi-asserted-by":"publisher","first-page":"2515","DOI":"10.1137\/19M1272512","article-title":"Domain decomposition with local impedance conditions for the Helmholtz equation with absorption","volume":"58","author":"Graham, Ivan G.","year":"2020","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"24","unstructured":"F. Hecht, Freefem++ manual (version 3.58-1), 2019."},{"issue":"8","key":"25","doi-asserted-by":"publisher","first-page":"1507","DOI":"10.1016\/j.cma.2006.03.016","article-title":"Restricted overlapping balancing domain decomposition methods and restricted coarse problems for the Helmholtz problem","volume":"196","author":"Kimn, Jung-Han","year":"2007","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"26","unstructured":"D. Lafontaine and E. A. Spence, Sharp bounds on Helmholtz impedance-to-impedance maps and application to overlapping domain decomposition, In preparation, 2021."},{"issue":"1","key":"27","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1007\/s00211-021-01253-0","article-title":"A sharp relative-error bound for the Helmholtz \u210e-FEM at high frequency","volume":"150","author":"Lafontaine, D.","year":"2022","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"28","isbn-type":"print","volume-title":"Strongly elliptic systems and boundary integral equations","author":"McLean, William","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0521663326"},{"key":"29","doi-asserted-by":"publisher","first-page":"113162","DOI":"10.1016\/j.cma.2020.113162","article-title":"A non-overlapping domain decomposition method with high-order transmission conditions and cross-point treatment for Helmholtz problems","volume":"368","author":"Modave, A.","year":"2020","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"30","unstructured":"O. R. Pembery, The Helmholtz equation in heterogeneous and random media: analysis and numerics. PhD Thesis, University of Bath, 2020, \\url{https:\/\/researchportal.bath.ac.uk\/en\/studentTheses\/the-helmholtz-equation-in-heterogeneous-and-random-media-analysis}."},{"issue":"2","key":"31","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/s00607-006-0177-z","article-title":"A refined finite element convergence theory for highly indefinite Helmholtz problems","volume":"78","author":"Sauter, S. A.","year":"2006","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"key":"32","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1201\/9780429507069","volume-title":"Variational techniques for elliptic partial differential equations","author":"Sayas, Francisco-Javier","year":"2019","ISBN":"https:\/\/id.crossref.org\/isbn\/9781138580886"},{"key":"33","doi-asserted-by":"publisher","first-page":"959","DOI":"10.2307\/2005357","article-title":"An observation concerning Ritz-Galerkin methods with indefinite bilinear forms","volume":"28","author":"Schatz, Alfred H.","year":"1974","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"190","key":"34","doi-asserted-by":"publisher","first-page":"483","DOI":"10.2307\/2008497","article-title":"Finite element interpolation of nonsmooth functions satisfying boundary conditions","volume":"54","author":"Scott, L. Ridgway","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"6","key":"35","doi-asserted-by":"publisher","first-page":"2402","DOI":"10.1137\/060652610","article-title":"Optimized multiplicative, additive, and restricted additive Schwarz preconditioning","volume":"29","author":"St-Cyr, A.","year":"2007","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"36","doi-asserted-by":"publisher","first-page":"109706","DOI":"10.1016\/j.jcp.2020.109706","article-title":"L-sweeps: a scalable, parallel preconditioner for the high-frequency Helmholtz equation","volume":"420","author":"Taus, Matthias","year":"2020","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"37","doi-asserted-by":"publisher","first-page":"88","DOI":"10.1016\/j.parco.2019.02.004","article-title":"Microwave tomographic imaging of cerebrovascular accidents by using high-performance computing","volume":"85","author":"Tournier, P.-H.","year":"2019","journal-title":"Parallel Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0167-8191","issn-type":"print"},{"issue":"3","key":"38","doi-asserted-by":"publisher","first-page":"1266","DOI":"10.1093\/imanum\/drt033","article-title":"Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I: linear version","volume":"34","author":"Wu, Haijun","year":"2014","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-339\/S0025-5718-2022-03772-8\/S0025-5718-2022-03772-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:45:51Z","timestamp":1776833151000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-339\/S0025-5718-2022-03772-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,6]]},"references-count":38,"journal-issue":{"issue":"339","published-print":{"date-parts":[[2023,1]]}},"alternative-id":["S0025-5718-2022-03772-8"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3772","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2022,10,6]]}}}