{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,15]],"date-time":"2025-08-15T02:06:31Z","timestamp":1755223591328,"version":"3.43.0"},"reference-count":40,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T00:00:00Z","timestamp":1742860800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12101588"],"award-info":[{"award-number":["12101588"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[2025,4]]},"abstract":"<jats:title>ABSTRACT<\/jats:title><jats:p>Block implicit method (BIM) is a family of time discretization schemes with desirable stability properties, high order of accuracy, and requires a single initial value. Moreover, BIM is also time\u2010parallel since the solutions at several time steps can be computed simultaneously at a time. However, its resulting linear system is often large and highly ill\u2010conditioned. In this paper, we present a multilevel additive Schwarz preconditioner for solving the large sparse system of algebraic equations arising from the discretization by BIM in time and finite element in space. Under some mild assumption, we prove an optimal convergence theory showing that the convergence rate is bounded independently of the spatial mesh sizes, the time step size, the number of subdomains, and the number of levels. Numerical experiments carried out on a parallel computer with thousands of processors confirm the optimality and scalability of the method, which performs better than the low\u2010order schemes in terms of the compute time.<\/jats:p>","DOI":"10.1002\/nla.70014","type":"journal-article","created":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T05:34:14Z","timestamp":1743053654000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Multilevel Additive Schwarz Preconditioners for High\u2010Order Block Implicit Methods for Parabolic Equations"],"prefix":"10.1002","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2163-9949","authenticated-orcid":false,"given":"Shishun","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics Xinyang Normal University  Henan P. R. China"}]},{"given":"Lei","family":"Xu","sequence":"additional","affiliation":[{"name":"Shenzhen Institutes of Advanced Technology Chinese Academy of Sciences  Shenzhen P. R. China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0296-8640","authenticated-orcid":false,"given":"Xiao\u2010Chuan","family":"Cai","sequence":"additional","affiliation":[{"name":"Department of Mathematics University of Macau  Macau P. R. 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