{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T21:04:29Z","timestamp":1767992669148,"version":"3.49.0"},"reference-count":16,"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>This paper addresses the challenge of proving the existence of solutions for nonlinear equations in Banach spaces,\nfocusing on the Navier\u2013Stokes equations and their discretizations.\nTraditional methods, such as monotonicity-based approaches and fixed-point theorems, often face limitations in handling general nonlinear operators\nor finite element discretizations.\nA novel concept, mapped coercivity, provides a unifying framework to analyze nonlinear operators through a continuous mapping.\nWe apply these ideas to saddle-point problems in Banach spaces, emphasizing both infinite-dimensional formulations and finite element discretizations.\nOur analysis includes stabilization techniques to restore coercivity in finite-dimensional settings, ensuring stability and existence of solutions.\nFor linear problems, we explore the relationship between the inf-sup condition and mapped coercivity, using the Stokes equation as a case study.\nFor nonlinear saddle-point systems, we extend the framework to mapped coercivity via surjective mappings,\nenabling concise proofs of existence of solutions for various stabilized Navier\u2013Stokes finite element methods.\nThese include Brezzi\u2013Pitk\u00e4ranta, a simple variant, and local projection stabilization (LPS) techniques, with extensions to convection-dominant flows.\nThe proposed methodology offers a robust tool for analyzing nonlinear PDEs and their discretizations,\nbypassing traditional decompositions and providing a foundation for future developments in computational fluid dynamics.<\/jats:p>","DOI":"10.1515\/cmam-2024-0187","type":"journal-article","created":{"date-parts":[[2025,5,4]],"date-time":"2025-05-04T10:00:32Z","timestamp":1746352832000},"page":"547-560","source":"Crossref","is-referenced-by-count":5,"title":["Mapped Coercivity for the Stationary Navier\u2013Stokes Equations and Their Finite Element Approximations"],"prefix":"10.1515","volume":"25","author":[{"given":"Roland","family":"Becker","sequence":"first","affiliation":[{"name":"Department of Mathematics , Universit\u00e9 de Pau et de l\u2019Adour (UPPA) , Avenue de l\u2019Universit\u00e9, BP 1155, 64013 Pau CEDEX , France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1730-6316","authenticated-orcid":false,"given":"Malte","family":"Braack","sequence":"additional","affiliation":[{"name":"Mathematical Seminar , 90299 Kiel University , Heinrich-Hecht-Platz 6, 24098 Kiel , Germany"}]}],"member":"374","published-online":{"date-parts":[[2025,4,30]]},"reference":[{"key":"2025070217063942119_j_cmam-2024-0187_ref_001","doi-asserted-by":"crossref","unstructured":"R.  Becker and M.  Braack,\nA finite element pressure gradient stabilization for the Stokes equations based on local projections,\nCalcolo 38 (2001), no. 4, 173\u2013199.","DOI":"10.1007\/s10092-001-8180-4"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_002","unstructured":"R.  Becker and M.  Braack,\nThe concept of mapped coercivity for nonlinear operators in Banach spaces,\npreprint (2024), https:\/\/arxiv.org\/abs\/2305.16783v2."},{"key":"2025070217063942119_j_cmam-2024-0187_ref_003","doi-asserted-by":"crossref","unstructured":"R.  Becker and M.  Braack,\nThe concept of mapped coercivity for nonlinear operators in Banach spaces,\nJ. Funct. Anal. 289 (2025), no. 3, Article ID 110893.","DOI":"10.1016\/j.jfa.2025.110893"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_004","doi-asserted-by":"crossref","unstructured":"R.  Becker and P.  Hansbo,\nA simple pressure stabilization method for the Stokes equation,\nComm. Numer. Methods Engrg. 24 (2008), no. 11, 1421\u20131430.","DOI":"10.1002\/cnm.1041"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_005","doi-asserted-by":"crossref","unstructured":"M.  Braack and E.  Burman,\nLocal projection stabilization for the Oseen problem and its interpretation as a variational multiscale method,\nSIAM J. Numer. Anal. 43 (2006), no. 6, 2544\u20132566.","DOI":"10.1137\/050631227"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_006","doi-asserted-by":"crossref","unstructured":"H.  Brezis,\n\u00c9quations et in\u00e9quations non lin\u00e9aires dans les espaces vectoriels en dualit\u00e9,\nAnn. Inst. Fourier (Grenoble) 18 (1968), 115\u2013175.","DOI":"10.5802\/aif.280"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_007","doi-asserted-by":"crossref","unstructured":"H.  Brezis,\nFunctional Analysis, Sobolev Spaces and Partial Differential Equations,\nUniversitext,\nSpringer, New York, 2011.","DOI":"10.1007\/978-0-387-70914-7"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_008","doi-asserted-by":"crossref","unstructured":"F.  Brezzi,\nOn the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers,\nRev. Fran\u00e7aise Automat. Informat. Rech. Op\u00e9r. S\u00e9r. Rouge 8 (1974), 129\u2013151.","DOI":"10.1051\/m2an\/197408R201291"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_009","doi-asserted-by":"crossref","unstructured":"F.  Brezzi and J.  Pitk\u00e4ranta,\nOn the stabilization of finite element approximations of the Stokes equations,\nEfficient Solutions of Elliptic Systems,\nVieweg & Teubner, Wiesbaden (1984), 11\u201319.","DOI":"10.1007\/978-3-663-14169-3_2"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_010","doi-asserted-by":"crossref","unstructured":"I.  Ekeland and R.  T\u00e9mam,\nConvex Analysis and Variational Problems,\nClassics Appl. Math. 28,\nSociety for Industrial and Applied Mathematics, Philadelphia, 1999.","DOI":"10.1137\/1.9781611971088"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_011","doi-asserted-by":"crossref","unstructured":"V.  Girault and P.-A.  Raviart,\nFinite Element Methods for Navier\u2013Stokes Equations,\nSpringer Ser. Comput. Math. 5,\nSpringer, Berlin, 1986.","DOI":"10.1007\/978-3-642-61623-5"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_012","doi-asserted-by":"crossref","unstructured":"R.  Glowinski,\nNumerical Methods for Nonlinear Variational Problems,\nSpringer Ser. Comput. Math.,\nSpringer, New York, 1984.","DOI":"10.1007\/978-3-662-12613-4"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_013","doi-asserted-by":"crossref","unstructured":"J.  Haslinger, M.  Miettinen and P. D.  Panagiotopoulos,\nFinite Element Method for Hemivariational Inequalities,\nNonconvex Optim. Appl. 35,\nKluwer Academic, Dordrecht, 1999.","DOI":"10.1007\/978-1-4757-5233-5"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_014","doi-asserted-by":"crossref","unstructured":"N.  Kenmochi,\nPseudomonotone operators and nonlinear elliptic boundary value problems,\nJ. Math. Soc. Japan 27 (1975), 121\u2013149.","DOI":"10.2969\/jmsj\/02710121"},{"key":"2025070217063942119_j_cmam-2024-0187_ref_015","unstructured":"J.-L.  Lions,\nQuelques m\u00e9thodes de r\u00e9solution des probl\u00e8mes aux limites non lin\u00e9aires,\nDunod, Paris, 1969."},{"key":"2025070217063942119_j_cmam-2024-0187_ref_016","doi-asserted-by":"crossref","unstructured":"L.  Tobiska and R.  Verf\u00fcrth,\nAnalysis of a streamline diffusion finite element method for the Stokes and Navier\u2013Stokes equations,\nSIAM J. Numer. Anal. 33 (1996), no. 1, 107\u2013127.","DOI":"10.1137\/0733007"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0187\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0187\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,2]],"date-time":"2025-07-02T17:08:20Z","timestamp":1751476100000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0187\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,30]]},"references-count":16,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,6,17]]},"published-print":{"date-parts":[[2025,7,1]]}},"alternative-id":["10.1515\/cmam-2024-0187"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2024-0187","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,30]]}}}