{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,25]],"date-time":"2025-10-25T00:33:53Z","timestamp":1761352433533,"version":"3.40.5"},"reference-count":28,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2024,2,28]],"date-time":"2024-02-28T00:00:00Z","timestamp":1709078400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["NRF-2021R1F1A106243413","NRF-2022R1A2B5B02002481","NRF-2020R1I1A1A01070361"],"award-info":[{"award-number":["NRF-2021R1F1A106243413","NRF-2022R1A2B5B02002481","NRF-2020R1I1A1A01070361"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems.\nWe have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton\u2019s method across different mesh levels, which is numerically validated.\nOur framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations.\nIn particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem.\nAs applications of our theory, we consider the pseudostress-velocity formulation of Navier\u2013Stokes equations and the standard Galerkin formulation of semilinear elliptic equations.\nReliable and efficient a posteriori error estimators for both approximations are derived.\nFinally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.<\/jats:p>","DOI":"10.1515\/cmam-2023-0088","type":"journal-article","created":{"date-parts":[[2024,2,27]],"date-time":"2024-02-27T19:52:49Z","timestamp":1709063569000},"page":"747-776","source":"Crossref","is-referenced-by-count":1,"title":["Adaptive Multi-level Algorithm for a Class of Nonlinear Problems"],"prefix":"10.1515","volume":"24","author":[{"given":"Dongho","family":"Kim","sequence":"first","affiliation":[{"name":"University College , 26721 Yonsei University , Seoul 03722 , Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3836-1595","authenticated-orcid":false,"given":"Eun-Jae","family":"Park","sequence":"additional","affiliation":[{"name":"Department of Computational Science and Engineering , 26721 Yonsei University , Seoul 03722 , Korea"}]},{"given":"Boyoon","family":"Seo","sequence":"additional","affiliation":[{"name":"Department of Mathematics , 26721 Yonsei University , Seoul 03722 , Korea"}]}],"member":"374","published-online":{"date-parts":[[2024,2,28]]},"reference":[{"key":"2024070116104898618_j_cmam-2023-0088_ref_001","doi-asserted-by":"crossref","unstructured":"M. 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Math. 36 (1980\/81), no. 1, 1\u201325.","DOI":"10.1007\/BF01395985"},{"key":"2024070116104898618_j_cmam-2023-0088_ref_005","doi-asserted-by":"crossref","unstructured":"Z. Cai, C. Wang and S. Zhang,\nMixed finite element methods for incompressible flow: Stationary Navier\u2013Stokes equations,\nSIAM J. Numer. Anal. 48 (2010), no. 1, 79\u201394.","DOI":"10.1137\/080718413"},{"key":"2024070116104898618_j_cmam-2023-0088_ref_006","doi-asserted-by":"crossref","unstructured":"Z. Cai and Y. Wang,\nA multigrid method for the pseudostress formulation of Stokes problems,\nSIAM J. Sci. Comput. 29 (2007), no. 5, 2078\u20132095.","DOI":"10.1137\/060661429"},{"key":"2024070116104898618_j_cmam-2023-0088_ref_007","doi-asserted-by":"crossref","unstructured":"G. Caloz and J. Rappaz,\nNumerical analysis for nonlinear and bifurcation problems,\nHandbook of Numerical Analysis. Vol. V,\nNorth-Holland, Amsterdam (1997), 487\u2013637.","DOI":"10.1016\/S1570-8659(97)80004-X"},{"key":"2024070116104898618_j_cmam-2023-0088_ref_008","doi-asserted-by":"crossref","unstructured":"C. Carstensen,\nA posteriori error estimate for the mixed finite element method,\nMath. Comp. 66 (1997), no. 218, 465\u2013476.","DOI":"10.1090\/S0025-5718-97-00837-5"},{"key":"2024070116104898618_j_cmam-2023-0088_ref_009","doi-asserted-by":"crossref","unstructured":"C. Carstensen and S. A. Funken,\nA posteriori error control in low-order finite element discretisations of incompressible stationary flow problems,\nMath. Comp. 70 (2001), no. 236, 1353\u20131381.","DOI":"10.1090\/S0025-5718-00-01264-3"},{"key":"2024070116104898618_j_cmam-2023-0088_ref_010","doi-asserted-by":"crossref","unstructured":"C. Carstensen, D. Kim and E.-J. Park,\nA priori and a posteriori pseudostress-velocity mixed finite element error analysis for the Stokes problem,\nSIAM J. Numer. Anal. 49 (2011), no. 6, 2501\u20132523.","DOI":"10.1137\/100816237"},{"key":"2024070116104898618_j_cmam-2023-0088_ref_011","doi-asserted-by":"crossref","unstructured":"P. Cl\u00e9ment,\nApproximation by finite element functions using local regularization,\nRAIRO Oper. Res. 9 (1975), no. R-2, 77\u201384.","DOI":"10.1051\/m2an\/197509R200771"},{"key":"2024070116104898618_j_cmam-2023-0088_ref_012","doi-asserted-by":"crossref","unstructured":"V. Ervin, W. Layton and J. Maubach,\nA posteriori error estimators for a two-level finite element method for the Navier\u2013Stokes equations,\nNumer. Methods Partial Differential Equations 12 (1996), no. 3, 333\u2013346.","DOI":"10.1002\/(SICI)1098-2426(199605)12:3<333::AID-NUM4>3.0.CO;2-P"},{"key":"2024070116104898618_j_cmam-2023-0088_ref_013","doi-asserted-by":"crossref","unstructured":"V. Girault and P.-A. Raviart,\nFinite Element Methods for Navier\u2013Stokes equations,\nSpringer Ser. Comput. 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