{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T20:49:17Z","timestamp":1759178957247,"version":"3.41.0"},"reference-count":28,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100004543","name":"China Scholarship Council","doi-asserted-by":"publisher","award":["202106380059"],"award-info":[{"award-number":["202106380059"]}],"id":[{"id":"10.13039\/501100004543","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["EXC 2181\/1 \u2013 390900948"],"award-info":[{"award-number":["EXC 2181\/1 \u2013 390900948"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100021856","name":"Ministero dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["P2022N5ZNP"],"award-info":[{"award-number":["P2022N5ZNP"]}],"id":[{"id":"10.13039\/501100021856","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009112","name":"Istituto Nazionale di Alta Matematica \u201dFrancesco Severi\u201d (INDAM) - Gruppo Nazionale per il Calcolo Scientifico","doi-asserted-by":"publisher","award":["E53C23001670001","E53C24001950001"],"award-info":[{"award-number":["E53C23001670001","E53C24001950001"]}],"id":[{"id":"10.13039\/100009112","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We discuss vertex patch smoothers as overlapping domain decomposition methods for fourth order elliptic partial differential equations. We show that they are numerically very efficient and yield high convergence rates. Furthermore, we discuss low rank tensor approximations for their efficient implementation. Our experiments demonstrate that the inexact local solver yields a method which converges fast and uniformly with respect to mesh refinement and polynomial degree. The multiplicative smoother shows superior performance in terms of solution efficiency, requiring fewer iterations in both two- and three-dimensional cases. Additionally, the solver infrastructure supports a mixed-precision approach, executing the multigrid preconditioner in single precision while performing the outer iteration in double precision, thereby increasing throughput by up to 70 %.<\/jats:p>","DOI":"10.1515\/cmam-2024-0192","type":"journal-article","created":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T18:07:46Z","timestamp":1748628466000},"page":"695-708","source":"Crossref","is-referenced-by-count":2,"title":["Tensor-Product Vertex Patch Smoothers for Biharmonic Problems"],"prefix":"10.1515","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7524-542X","authenticated-orcid":false,"given":"Julius","family":"Witte","sequence":"first","affiliation":[{"name":"Interdisciplinary Center for Scientific Computing (IWR) , 9144 Heidelberg University , Im Neuenheimer Feld 205, 69120 Heidelberg , Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0341-4447","authenticated-orcid":false,"given":"Cu","family":"Cui","sequence":"additional","affiliation":[{"name":"Interdisciplinary Center for Scientific Computing (IWR) , 9144 Heidelberg University , Im Neuenheimer Feld 205, 69120 Heidelberg , Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6222-3352","authenticated-orcid":false,"given":"Francesca","family":"Bonizzoni","sequence":"additional","affiliation":[{"name":"MOX-Department of Mathematics , Politecnico di Milano , Piazza Leonardo da Vinci 32, 20133 Milano , Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1687-7328","authenticated-orcid":false,"given":"Guido","family":"Kanschat","sequence":"additional","affiliation":[{"name":"Interdisciplinary Center for Scientific Computing (IWR) , 9144 Heidelberg University , Im Neuenheimer Feld 205, 69120 Heidelberg , Germany"}]}],"member":"374","published-online":{"date-parts":[[2025,5,29]]},"reference":[{"key":"2025070217063876281_j_cmam-2024-0192_ref_001","doi-asserted-by":"crossref","unstructured":"D.  Arndt, W.  Bangerth, M.  Bergbauer, M.  Feder, M.  Fehling, J.  Heinz, T.  Heister, L.  Heltai, M.  Kronbichler, M.  Maier, P.  Munch, J.-P.  Pelteret, B.  Turcksin, D.  Wells and S.  Zampini,\nThe deal.II library, version 9.5,\nJ. Numer. Math. 31 (2023), no. 3, 231\u2013246.","DOI":"10.1515\/jnma-2023-0089"},{"key":"2025070217063876281_j_cmam-2024-0192_ref_002","doi-asserted-by":"crossref","unstructured":"D. N.  Arnold,\nAn interior penalty finite element method with discontinuous elements,\nSIAM J. Numer. Anal. 19 (1982), no. 4, 742\u2013760.","DOI":"10.1137\/0719052"},{"key":"2025070217063876281_j_cmam-2024-0192_ref_003","doi-asserted-by":"crossref","unstructured":"D. N.  Arnold, R. S.  Falk and R.  Winther,\nPreconditioning in \n                  \n                     \n                        \n                           H\n                           \u2062\n                           \n                              (\n                              div\n                              )\n                           \n                        \n                     \n                     \n                     {H({\\rm div})}\n                  \n                and applications,\nMath. Comp. 66 (1997), no. 219, 957\u2013984.","DOI":"10.1090\/S0025-5718-97-00826-0"},{"key":"2025070217063876281_j_cmam-2024-0192_ref_004","doi-asserted-by":"crossref","unstructured":"S. C.  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Comput. 22\/23 (2005), 83\u2013118.","DOI":"10.1007\/s10915-004-4135-7"},{"key":"2025070217063876281_j_cmam-2024-0192_ref_006","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and K.  Wang,\nTwo-level additive Schwarz preconditioners for \n                  \n                     \n                        \n                           C\n                           0\n                        \n                     \n                     \n                     {C^{0}}\n                  \n                interior penalty methods,\nNumer. Math. 102 (2005), no. 2, 231\u2013255.","DOI":"10.1007\/s00211-005-0641-2"},{"key":"2025070217063876281_j_cmam-2024-0192_ref_007","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and J.  Zhao,\nConvergence of multigrid algorithms for interior penalty methods,\nAppl. Numer. Anal. Comput. 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Zhang,\nAn optimal order multigrid method for biharmonic, \n                  \n                     \n                        \n                           C\n                           1\n                        \n                     \n                     \n                     {C^{1}}\n                  \n                finite element equations,\nNumer. Math. 56 (1989), no. 6, 613\u2013624.","DOI":"10.1007\/BF01396347"},{"key":"2025070217063876281_j_cmam-2024-0192_ref_027","doi-asserted-by":"crossref","unstructured":"J.  Zhao,\nConvergence of V- and F-cycle multigrid methods for the biharmonic problem using the Hsieh\u2013Clough\u2013Tocher element,\nNumer. Methods Partial Differential Equations 21 (2005), no. 3, 451\u2013471.","DOI":"10.1002\/num.20048"},{"key":"2025070217063876281_j_cmam-2024-0192_ref_028","unstructured":"NVIDIA Corporation, Nsight compute, 2023."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0192\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0192\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,2]],"date-time":"2025-07-02T17:07:05Z","timestamp":1751476025000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0192\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,29]]},"references-count":28,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,6,17]]},"published-print":{"date-parts":[[2025,7,1]]}},"alternative-id":["10.1515\/cmam-2024-0192"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2024-0192","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2025,5,29]]}}}