{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T13:38:24Z","timestamp":1762349904638,"version":"build-2065373602"},"reference-count":34,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12288101","12301466"],"award-info":[{"award-number":["12288101","12301466"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>A new method is proposed to provide guaranteed lower bounds for eigenvalues of general second order elliptic\noperators in any dimension. This method employs a novel generalized Crouzeix\u2013Raviart element which is proved to yield asymptotic lower bounds for eigenvalues of general second order elliptic operators, and a simple post-processing method.\nAs a byproduct, a simple and cheap method is also proposed to obtain guaranteed upper bounds for eigenvalues, which is based\non generalized Crouzeix\u2013Raviart element approximate eigenfunctions, an averaging interpolation from the generalized Crouzeix\u2013Raviart element space to the conforming linear element space, and an usual Rayleigh\u2013Ritz procedure.\nThe ingredients for the analysis consist of a crucial projection property of the canonical interpolation operator of the generalized Crouzeix\u2013Raviart element,\nexplicitly computable constants for two interpolation operators. Numerical experiments demonstrate that the guaranteed lower bounds for eigenvalues in this paper are superior to those obtained by the Crouzeix\u2013Raviart element.<\/jats:p>","DOI":"10.1515\/cmam-2024-0168","type":"journal-article","created":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T16:39:22Z","timestamp":1748623162000},"page":"863-881","source":"Crossref","is-referenced-by-count":0,"title":["Guaranteed Lower and Upper Bounds for Eigenvalues of Second Order Elliptic Operators in any Dimension"],"prefix":"10.1515","volume":"25","author":[{"given":"Jun","family":"Hu","sequence":"first","affiliation":[{"name":"LMAM and School of Mathematical Sciences , 12465 Peking University , Beijing 100871 , P. R. China"}]},{"given":"Rui","family":"Ma","sequence":"additional","affiliation":[{"name":"Beijing Institute of Technology , Beijing 100081 , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2025,5,29]]},"reference":[{"key":"2025110513331515936_j_cmam-2024-0168_ref_001","unstructured":"M. G.  Armentano and R. G.  Dur\u00e1n,\nAsymptotic lower bounds for eigenvalues by nonconforming finite element methods,\nElectron. Trans. Numer. Anal. 17 (2004), 93\u2013101."},{"key":"2025110513331515936_j_cmam-2024-0168_ref_002","doi-asserted-by":"crossref","unstructured":"I.  Babu\u0161ka and J.  Osborn,\nEigenvalue problems,\nHandbook of Numerical Analysis, Vol. II,\nNorth-Holland, Amsterdam (1991), 641\u2013787.","DOI":"10.1016\/S1570-8659(05)80042-0"},{"key":"2025110513331515936_j_cmam-2024-0168_ref_003","doi-asserted-by":"crossref","unstructured":"D.  Boffi,\nFinite element approximation of eigenvalue problems,\nActa Numer. 19 (2010), 1\u2013120.","DOI":"10.1017\/S0962492910000012"},{"key":"2025110513331515936_j_cmam-2024-0168_ref_004","doi-asserted-by":"crossref","unstructured":"C.  Carstensen, A.  Ern and S.  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Sin. 29 (2007), no. 3, 319\u2013321."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0168\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0168\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T13:35:27Z","timestamp":1762349727000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0168\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,29]]},"references-count":34,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,6,25]]},"published-print":{"date-parts":[[2025,10,1]]}},"alternative-id":["10.1515\/cmam-2024-0168"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2024-0168","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]]}}}