{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T15:15:19Z","timestamp":1760886919695,"version":"3.43.0"},"reference-count":16,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[1999,8,1]],"date-time":"1999-08-01T00:00:00Z","timestamp":933465600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[1999,8,1]],"date-time":"1999-08-01T00:00:00Z","timestamp":933465600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Reliable Computing"],"published-print":{"date-parts":[[1999,8]]},"DOI":"10.1023\/a:1009976505460","type":"journal-article","created":{"date-parts":[[2002,12,22]],"date-time":"2002-12-22T19:25:11Z","timestamp":1040585111000},"page":"347-357","source":"Crossref","is-referenced-by-count":11,"title":["A Numerical Verification Method of Solutions for the Navier-Stokes Equations"],"prefix":"10.1007","volume":"5","author":[{"given":"Yoshitaka","family":"Watanabe","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nobito","family":"Yamamoto","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mitsuhiro T.","family":"Nakao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","reference":[{"key":"244308_CR1","volume-title":"The Finite Element Method for Elliptic Problems","author":"P. G. Ciarlet","year":"1978","unstructured":"Ciarlet, P. G.: The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978."},{"key":"244308_CR2","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61623-5","volume-title":"Finite Element Approximation of the Navier-Stokes Equations","author":"V. Girault","year":"1986","unstructured":"Girault, V. and Raviart, P. A.: Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, Berlin, Heidelberg, 1986."},{"key":"244308_CR3","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4613-3440-8","volume-title":"Applications of Interval Computations","author":"R. B. Kearfott","year":"1996","unstructured":"Kearfott, R. B. and Kreinovich, V.: Applications of Interval Computations, Kluwer Academic Publishers, Netherlands, 1996."},{"key":"244308_CR4","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/BF02307379","volume":"53","author":"O. Kn\u00fcppel","year":"1994","unstructured":"Kn\u00fcppel, O.: PROFIL\/BIAS\u2014A Fast Interval Library, Computing\n53 (1994), pp. 277\u2013287.","journal-title":"Computing"},{"key":"244308_CR5","volume-title":"C-XSC, A C++ Class Library for Extended Scientific Computing","author":"U. Kulisch","year":"1993","unstructured":"Kulisch, U. et al.: C-XSC, A C++ Class Library for Extended Scientific Computing, Springer-Verlag, New York, 1993."},{"key":"244308_CR6","volume-title":"The Mathematical Theory of Viscous Incompressible Flow","author":"O. Ladyzhenskaya","year":"1969","unstructured":"Ladyzhenskaya, O.: The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York, 1969."},{"key":"244308_CR7","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/0022-247X(92)90129-2","volume":"164","author":"M. T. Nakao","year":"1992","unstructured":"Nakao, M. T.: A Numerical Verification Method for the Existence of Weak Solutions for Nonlinear Boundary Value Problems, J. Math. Anal. Appl.\n164 (1992), pp. 489\u2013507.","journal-title":"J. Math. Anal. Appl."},{"key":"244308_CR8","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/978-3-7091-6918-6_13","volume":"9","author":"M. T. Nakao","year":"1993","unstructured":"Nakao M. T.: Solving Nonlinear Elliptic Problems with Result Verification Using an H\n\u22121 Type Residual Iteration, Computing, Suppl.\n9 (1993), pp. 161\u2013173.","journal-title":"Computing, Suppl."},{"key":"244308_CR9","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1006\/jath.1998.3172","volume":"93","author":"M. T. Nakao","year":"1998","unstructured":"Nakao, M. T., Yamamoto, N., and Kimura, S.: On Best Constant in the Optimal Error Estimates for the H\n10-projection into Piecewise Polynomial Spaces, J. Approximation Theory\n93 (1998), pp. 491\u2013500.","journal-title":"J. Approximation Theory"},{"key":"244308_CR10","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/S0377-0427(98)00035-1","volume":"91","author":"M. T. Nakao","year":"1998","unstructured":"Nakao, M. T., Yamamoto, N., and Watanabe, Y.: A Posteriori and Constructive A Priori Error Bounds for Finite Element Solutions of the Stokes Equations, J. Comput. Appl. Math.\n91 (1998), pp. 137\u2013158.","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"244308_CR11","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1023\/A:1009979408417","volume":"4","author":"M. T. Nakao","year":"1998","unstructured":"Nakao, M. T., Yamamoto, N., and Watanabe, Y.: Constructive L\n2 Error Estimates for Finite Element Solutions of the Stokes Equations, Reliable Computing\n4(2) (1998), pp. 115\u2013124.","journal-title":"Reliable Computing"},{"key":"244308_CR12","unstructured":"Nakao, M. T., Yamamoto, N., and Watanabe, Y.: Guaranteed Error Bounds for the Finite Element Solutions of the Stokes Problem, in: Alefeld, G., Frommer, A., and Lang, B. (eds): Scientific Computing and Validated Numerics, Proceedings of International Symposium on Scientific Computing, Computing Arithmetic and Validated Numerics SCAN-95, Mathematical Research\n90 (1996), pp. 258\u2013264."},{"key":"244308_CR13","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1007\/BF02320201","volume":"47","author":"S. M. Rump","year":"1992","unstructured":"Rump, S. M.: On the Solution of Interval Linear Systems, Computing\n47 (1992), pp. 337\u2013353.","journal-title":"Computing"},{"key":"244308_CR14","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1007\/BF02418013","volume":"110","author":"G. Talenti","year":"1976","unstructured":"Talenti, G.: Best Constant in Sobolev Inequality, Ann. Math. Pure Appl.\n110 (1976), pp 353\u2013372.","journal-title":"Ann. Math. Pure Appl."},{"key":"244308_CR15","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/BF03167208","volume":"10","author":"Y. Watanabe","year":"1993","unstructured":"Watanabe, Y., and Nakao, M. T.: Numerical Verifications of Solutions for Nonlinear Elliptic Equations, Japan J. Indust. Appl. Math.\n10 (1993) pp. 165\u2013178.","journal-title":"Japan J. Indust. Appl. Math."},{"key":"244308_CR16","doi-asserted-by":"crossref","first-page":"2004","DOI":"10.1137\/S0036142996304498","volume":"35","author":"N. Yamamoto","year":"1998","unstructured":"Yamamoto, N.: A Numerical Verification Method for Solutions of Boundary Value Problems with Local Uniqueness by Banach's Fixed-point Theorem, SIAM J. Numer. Anal.\n35 (1998) pp. 2004\u20132013.","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Reliable Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1023\/A:1009976505460.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1023\/A:1009976505460\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1023\/A:1009976505460.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T09:35:59Z","timestamp":1754472959000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1023\/A:1009976505460"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,8]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1999,8]]}},"alternative-id":["244308"],"URL":"https:\/\/doi.org\/10.1023\/a:1009976505460","relation":{},"ISSN":["1385-3139","1573-1340"],"issn-type":[{"type":"print","value":"1385-3139"},{"type":"electronic","value":"1573-1340"}],"subject":[],"published":{"date-parts":[[1999,8]]}}}