{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T01:01:26Z","timestamp":1768525286818,"version":"3.49.0"},"reference-count":35,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100002261","name":"Russian Foundation for Basic Research","doi-asserted-by":"publisher","award":["18-41-160014"],"award-info":[{"award-number":["18-41-160014"]}],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002261","name":"Russian Foundation for Basic Research","doi-asserted-by":"publisher","award":["19-08-01184"],"award-info":[{"award-number":["19-08-01184"]}],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>A class of Lagrangian mixed finite element methods is constructed for an approximate solution of a problem of nonlinear thin elastic shell theory, namely, the problem of finding critical points of the functional of potential energy according to the Budiansky\u2013Sanders model.\nThe proposed numerical method is based on the use of the second derivatives\nof the deflection as auxiliary variables.\nSufficient conditions for the solvability of the corresponding discrete problem\nare obtained. Accuracy estimates for approximate solutions are established.\nIterative methods for solving the corresponding systems of nonlinear equations are\nproposed and investigated.<\/jats:p>","DOI":"10.1515\/cmam-2019-0017","type":"journal-article","created":{"date-parts":[[2019,6,14]],"date-time":"2019-06-14T09:10:56Z","timestamp":1560503456000},"page":"631-642","source":"Crossref","is-referenced-by-count":1,"title":["Lagrangian Mixed Finite Element Methods for Nonlinear Thin Shell Problems"],"prefix":"10.1515","volume":"20","author":[{"given":"Mikhail M.","family":"Karchevsky","sequence":"first","affiliation":[{"name":"Department of Numerical Mathematics , Kazan Federal University , 18 Kremliovskaya Street , Kazan , 420008 , Russia"}]}],"member":"374","published-online":{"date-parts":[[2019,6,13]]},"reference":[{"key":"2023033110450942861_j_cmam-2019-0017_ref_001","doi-asserted-by":"crossref","unstructured":"G. P.  Astrakhantsev,\nOn a mixed finite element method in problems in the theory of shells,\nComput. Math. Math. Phys 29 (1989), no. 5, 167\u2013176.","DOI":"10.1016\/0041-5553(89)90195-X"},{"key":"2023033110450942861_j_cmam-2019-0017_ref_002","doi-asserted-by":"crossref","unstructured":"M.  Bernadou, P. G.  Ciarlet and B.  Miara,\nExistence theorems for two-dimensional linear shell theories,\nJ. Elasticity 34 (1994), no. 2, 111\u2013138.","DOI":"10.1007\/BF00041188"},{"key":"2023033110450942861_j_cmam-2019-0017_ref_003","doi-asserted-by":"crossref","unstructured":"C.  Bernardi,\nOptimal finite-element interpolation on curved domains,\nSIAM J. Numer. Anal. 26 (1989), no. 5, 1212\u20131240.","DOI":"10.1137\/0726068"},{"key":"2023033110450942861_j_cmam-2019-0017_ref_004","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":"2023033110450942861_j_cmam-2019-0017_ref_005","unstructured":"P. G.  Ciarlet,\nMathematical Elasticity. Vol. III: Theory of Shells,\nStud. Math. Appl. 29,\nNorth-Holland, Amsterdam, 2000."},{"key":"2023033110450942861_j_cmam-2019-0017_ref_006","doi-asserted-by":"crossref","unstructured":"P. G.  Ciarlet,\nThe Finite Element Method for Elliptic Problems,\nClass. Appl. Math. 40,\nSociety for Industrial and Applied Mathematics (SIAM), Philadelphia, 2002.","DOI":"10.1137\/1.9780898719208"},{"key":"2023033110450942861_j_cmam-2019-0017_ref_007","unstructured":"R. Z.  Dautov and M. M.  Karchevsky,\nIntroduction to the Theory of Finite Element Method (in Russian),\nKazan State University, Kazan, 2011."},{"key":"2023033110450942861_j_cmam-2019-0017_ref_008","doi-asserted-by":"crossref","unstructured":"P.  Destuynder,\nOn nonlinear membrane theory,\nComput. Methods Appl. Mech. Engrg. 32 (1982), no. 1\u20133, 377\u2013399.","DOI":"10.1016\/0045-7825(82)90077-9"},{"key":"2023033110450942861_j_cmam-2019-0017_ref_009","doi-asserted-by":"crossref","unstructured":"P.  Destuynder,\nAn existence theorem for a nonlinear shell model in large displacements analysis,\nMath. Methods Appl. Sci. 5 (1983), no. 1, 68\u201383.","DOI":"10.1002\/mma.1670050106"},{"key":"2023033110450942861_j_cmam-2019-0017_ref_010","unstructured":"K.  Hellan,\nAnalysis of elastic plates in flexure by a simplified finite element method,\nTechnical Report Civ. Eng. Series 46, Acta Polytechnica Scandinavia, Trondheim, 1967."},{"key":"2023033110450942861_j_cmam-2019-0017_ref_011","doi-asserted-by":"crossref","unstructured":"L. R.  Herrman,\nFinite element bending analysis of plates,\nJ. Eng. Mech. Div. 93 (1967), no. 5, 13\u201326.","DOI":"10.1061\/JMCEA3.0000891"},{"key":"2023033110450942861_j_cmam-2019-0017_ref_012","doi-asserted-by":"crossref","unstructured":"C.  Johnson,\nOn the convergence of a mixed finite-element method for plate bending problems,\nNumer. Math. 21 (1973), 43\u201362.","DOI":"10.1007\/BF01436186"},{"key":"2023033110450942861_j_cmam-2019-0017_ref_013","unstructured":"L. V.  Kantorovich and G. P.  Akilov,\nFunctional Analysis, 2nd ed.,\nPergamon Press, Oxford, 1982."},{"key":"2023033110450942861_j_cmam-2019-0017_ref_014","unstructured":"M. M.  Karchevsky,\nA mixed finite element method for nonlinear problems in the theory of plates,\nRuss. Math. 36 (1992), no. 7, 10\u201317."},{"key":"2023033110450942861_j_cmam-2019-0017_ref_015","unstructured":"M. M.  Karchevsky,\nOn the mixed finite-element method in the nonlinear theory of thin shells,\nZh. Vychisl. Mat. Mat. Fiz. 38 (1998), no. 2, 324\u2013329."},{"key":"2023033110450942861_j_cmam-2019-0017_ref_016","unstructured":"M. M.  Karchevsky,\nOn a class of difference schemes for nonlinear problems in the theory of plates,\nZh. Vychisl. Mat. Mat. 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