{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,3]],"date-time":"2025-07-03T04:13:33Z","timestamp":1751516013968,"version":"3.41.0"},"reference-count":34,"publisher":"Walter de Gruyter GmbH","issue":"3","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>In this paper, we observe an interesting phenomenon for a hybridizable discontinuous Galerkin (HDG) method for eigenvalue problems. Specifically, using the same finite element method, we may achieve both upper and lower eigenvalue bounds simultaneously, simply by the fine tuning of the stabilization parameter. Based on this observation, a high accuracy algorithm for computing eigenvalues is designed to yield higher convergence rate at a lower computational cost. Meanwhile, we demonstrate that certain type of HDG methods can only provide upper bounds. As a by-product, the asymptotic upper bound property of the Brezzi\u2013Douglas\u2013Marini mixed finite element is also established. Numerical results supporting our theory are given.<\/jats:p>","DOI":"10.1515\/cmam-2024-0193","type":"journal-article","created":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T20:42:23Z","timestamp":1748637743000},"page":"643-663","source":"Crossref","is-referenced-by-count":1,"title":["Computing Both Upper and Lower Eigenvalue Bounds by HDG Methods"],"prefix":"10.1515","volume":"25","author":[{"given":"Qigang","family":"Liang","sequence":"first","affiliation":[{"name":"School of Mathematical Science , Tongji University ; and Key Laboratory of Intelligent Computing and Applications, Tongji University, Ministry of Education , Shanghai 200092 , P. R. China"}]},{"given":"Xuejun","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematical Science , Tongji University ; and Key Laboratory of Intelligent Computing and Applications, Tongji University, Ministry of Education , Shanghai 200092 , P. R. China"}]},{"given":"Liuyao","family":"Yuan","sequence":"additional","affiliation":[{"name":"School of Mathematical Science , Tongji University ; and Key Laboratory of Intelligent Computing and Applications, Tongji University, Ministry of Education , Shanghai 200092 , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2025,5,29]]},"reference":[{"key":"2025070217063967274_j_cmam-2024-0193_ref_001","unstructured":"R. A.  Adams,\nSobolev Spaces,\nPure Appl. Math. 65,\nAcademic Press, New York, 1975."},{"key":"2025070217063967274_j_cmam-2024-0193_ref_002","doi-asserted-by":"crossref","unstructured":"P. F.  Antonietti, A.  Buffa and I.  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Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2025070217063967274_j_cmam-2024-0193_ref_006","doi-asserted-by":"crossref","unstructured":"F.  Brezzi, J.  Douglas, Jr. and L. D.  Marini,\nTwo families of mixed finite elements for second order elliptic problems,\nNumer. Math. 47 (1985), no. 2, 217\u2013235.","DOI":"10.1007\/BF01389710"},{"key":"2025070217063967274_j_cmam-2024-0193_ref_007","doi-asserted-by":"crossref","unstructured":"F.  Brezzi and M.  Fortin,\nMixed and Hybrid Finite Element Methods,\nSpringer Ser. Comput. Math. 15,\nSpringer, New York, 1991.","DOI":"10.1007\/978-1-4612-3172-1"},{"key":"2025070217063967274_j_cmam-2024-0193_ref_008","doi-asserted-by":"crossref","unstructured":"C.  Carstensen, A.  Ern and S.  Puttkammer,\nGuaranteed lower bounds on eigenvalues of elliptic operators with a hybrid high-order method,\nNumer. 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