{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T17:21:17Z","timestamp":1758820877526,"version":"3.41.0"},"reference-count":28,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100006443","name":"Vysok\u00e1 \u0160kola b\u00e1nsk\u00e1 - Technick\u00e1 Univerzita Ostrava","doi-asserted-by":"publisher","award":["SP2023\/67"],"award-info":[{"award-number":["SP2023\/67"]}],"id":[{"id":"10.13039\/501100006443","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001823","name":"Ministerstvo \u0160kolstv\u00ed, Ml\u00e1de\u017ee a T\u011blov\u00fdchovy","doi-asserted-by":"publisher","award":["90140"],"award-info":[{"award-number":["90140"]}],"id":[{"id":"10.13039\/501100001823","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We present a recently developed preconditioning of square block matrices (PRESB) to be used within a parallel method to solve linear systems of equations arising from tensor-product discretizations of initial boundary-value problems for parabolic partial differential equations.\nWe consider weak formulations in Bochner\u2013Sobolev spaces and tensor-product finite element approximations for the heat and eddy current equations.\nThe fast diagonalization method is employed to decouple the arising linear system of equations into auxiliary spatial complex-valued linear systems that can be solved concurrently.\nWe prove that the real part of the system matrix is positive definite, which allows us to accelerate the flexible generalized minimal residual method (FGMRES) by the PRESB preconditioner.\nThe action of PRESB on a vector includes two solutions with positive definite matrices.\nThe spectrum of the preconditioned system lies between 1\/2 and 1.\nFinally, we combine the PRESB-FGMRES method with multigrid-CG iterations and illustrate the numerical efficiency and the robustness for spatial discretizations up to 12 millions degrees of freedom.<\/jats:p>","DOI":"10.1515\/cmam-2023-0085","type":"journal-article","created":{"date-parts":[[2024,1,24]],"date-time":"2024-01-24T18:40:51Z","timestamp":1706121651000},"page":"439-451","source":"Crossref","is-referenced-by-count":2,"title":["Robust PRESB Preconditioning of a 3-Dimensional Space-Time Finite Element Method for Parabolic Problems"],"prefix":"10.1515","volume":"24","author":[{"given":"Ladislav","family":"Foltyn","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics , Faculty of Electrical Engineering and Computer Science , V\u0160B-Technical University of Ostrava , Ostrava , Czech Republic ; and IT4Innovations"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5818-4362","authenticated-orcid":false,"given":"Dalibor","family":"Luk\u00e1\u0161","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics , Faculty of Electrical Engineering and Computer Science , V\u0160B-Technical University of Ostrava , Ostrava , Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4146-1474","authenticated-orcid":false,"given":"Marco","family":"Zank","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Angewandte Mathematik , TU Graz , Graz , Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2024,1,25]]},"reference":[{"key":"2025062712372284730_j_cmam-2023-0085_ref_001","doi-asserted-by":"crossref","unstructured":"D. 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