{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T07:32:27Z","timestamp":1768548747023,"version":"3.49.0"},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100003443","name":"Ministry of Education and Science of the Russian Federation","doi-asserted-by":"publisher","award":["02.a03.21.0008"],"award-info":[{"award-number":["02.a03.21.0008"]}],"id":[{"id":"10.13039\/501100003443","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><jats:p>In numerical solving boundary value problems for parabolic equations, two- or three-level implicit schemes are in common use. Their computational implementation is based on solving a discrete elliptic problem at a new time level. For this purpose, various iterative methods are applied to multidimensional problems evaluating an approximate solution with some error. It is necessary to ensure that these errors do not violate the stability of the approximate solution, i.e., the approximate solution must converge to the exact one. In the present paper, these questions are investigated in numerical solving a Cauchy problem for a linear evolutionary equation of first order, which is considered in a finite-dimensional Hilbert space. The study is based on the general theory of stability (well-posedness) of operator-difference schemes developed by Samarskii. The iterative methods used in the study are considered from the same general positions.<\/jats:p>","DOI":"10.1515\/cmam-2018-0295","type":"journal-article","created":{"date-parts":[[2019,3,12]],"date-time":"2019-03-12T09:01:03Z","timestamp":1552381263000},"page":"727-737","source":"Crossref","is-referenced-by-count":3,"title":["Incomplete Iterative Implicit Schemes"],"prefix":"10.1515","volume":"20","author":[{"given":"Petr N.","family":"Vabishchevich","sequence":"first","affiliation":[{"name":"Nuclear Safety Institute , Russian Academy of Sciences, 52, B. Tulskaya, 115191; and Peoples\u2019 Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, 117198 Moscow , Russia"}]}],"member":"374","published-online":{"date-parts":[[2019,3,12]]},"reference":[{"key":"2023033110450796913_j_cmam-2018-0295_ref_001","doi-asserted-by":"crossref","unstructured":"U. M. Ascher, Numerical Methods for Evolutionary Differential Equations, Comput. Sci. Eng. 5, Society for Industrial and Applied Mathematics, Philadelphia, 2008.","DOI":"10.1137\/1.9780898718911"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_002","doi-asserted-by":"crossref","unstructured":"J. H. Bramble, J. E. Pasciak, P. H. Sammon and V. Thom\u00e9e, Incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data, Math. Comp. 52 (1989), no. 186, 339\u2013367.","DOI":"10.1090\/S0025-5718-1989-0962207-8"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_003","doi-asserted-by":"crossref","unstructured":"J. H. Bramble and P. H. Sammon, Efficient higher order single step methods for parabolic problems. I, Math. Comp. 35 (1980), no. 151, 655\u2013677.","DOI":"10.1090\/S0025-5718-1980-0572848-X"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_004","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Springer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_005","doi-asserted-by":"crossref","unstructured":"T. A. Davis, Direct Methods for Sparse Linear Systems, Fundam. Algorithms 2, Society for Industrial and Applied Mathematics, Philadelphia, 2006.","DOI":"10.1137\/1.9780898718881"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_006","doi-asserted-by":"crossref","unstructured":"J. Douglas, Jr., On incomplete iteration for implicit parabolic difference equations, J. Soc. Indust. Appl. Math. 9 (1961), 433\u2013439.","DOI":"10.1137\/0109036"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_007","doi-asserted-by":"crossref","unstructured":"J. Douglas, Jr., T. Dupont and R. E. Ewing, Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem, SIAM J. Numer. Anal. 16 (1979), no. 3, 503\u2013522.","DOI":"10.1137\/0716039"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_008","doi-asserted-by":"crossref","unstructured":"K. Gustafsson and G. S\u00f6derlind, Control strategies for the iterative solution of nonlinear equations in ODE solvers, SIAM J. Sci. Comput. 18 (1997), no. 1, 23\u201340; Dedicated to C. William Gear on the occasion of his 60th birthday.","DOI":"10.1137\/S1064827595287109"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_009","unstructured":"L. A. Hageman and D. M. Young, Applied Iterative Methods. Academic Press, New York, 1981."},{"key":"2023033110450796913_j_cmam-2018-0295_ref_010","doi-asserted-by":"crossref","unstructured":"W. Hundsdorfer and J. Verwer, Numerical Solution of Time-dependent Advection-Diffusion-Reaction Equations, Springer Ser. Comput. Math. 33, Springer, Berlin, 2003.","DOI":"10.1007\/978-3-662-09017-6"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_011","doi-asserted-by":"crossref","unstructured":"L. O. Jay, Inexact simplified Newton iterations for implicit Runge\u2013Kutta methods, SIAM J. Numer. Anal. 38 (2000), no. 4, 1369\u20131388.","DOI":"10.1137\/S0036142999360573"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_012","unstructured":"P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Texts Appl. Math. 44, Springer, New York, 2003."},{"key":"2023033110450796913_j_cmam-2018-0295_ref_013","unstructured":"K. W. Morton, Numerical Solution of Convection-diffusion Problems, Appl. Math. Math. Comput. 12, Chapman & Hall, London, 1996."},{"key":"2023033110450796913_j_cmam-2018-0295_ref_014","doi-asserted-by":"crossref","unstructured":"A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math. 23, Springer, Berlin, 1994.","DOI":"10.1007\/978-3-540-85268-1"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_015","doi-asserted-by":"crossref","unstructured":"Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, 2003.","DOI":"10.1137\/1.9780898718003"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_016","doi-asserted-by":"crossref","unstructured":"A. A. Samarskii, The Theory of Difference Schemes, Monogr. Textb. Pure Appl. Math. 240, Marcel Dekker, New York, 2001.","DOI":"10.1201\/9780203908518"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_017","doi-asserted-by":"crossref","unstructured":"A. A. Samarskii, P. P. Matus and P. N. Vabishchevich, Difference Schemes with Operator Factors, Math. Appl. 546, Kluwer Academic, Dordrecht, 2002.","DOI":"10.1007\/978-94-015-9874-3"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_018","unstructured":"A. A. Samarskii and A. V. Gulin, Stability of Difference Schemes (in Russian), Nauka, Moscow, 1973."},{"key":"2023033110450796913_j_cmam-2018-0295_ref_019","doi-asserted-by":"crossref","unstructured":"A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations. Vol. II. Iterative Methods, Birkh\u00e4user, Basel, 1989.","DOI":"10.1007\/978-3-0348-9272-8"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_020","unstructured":"A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Convection-Diffusion Problems (in Russian), URSS, Moscow, 1999."},{"key":"2023033110450796913_j_cmam-2018-0295_ref_021","doi-asserted-by":"crossref","unstructured":"J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, 2004.","DOI":"10.1137\/1.9780898717938"},{"key":"2023033110450796913_j_cmam-2018-0295_ref_022","unstructured":"V. Thom\u00e9e, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed., Springer Ser. Comput. Math. 25, Springer, Berlin, 2006."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/20\/4\/article-p727.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0295\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0295\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,15]],"date-time":"2024-07-15T18:32:23Z","timestamp":1721068343000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0295\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,12]]},"references-count":22,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,8,5]]},"published-print":{"date-parts":[[2020,10,1]]}},"alternative-id":["10.1515\/cmam-2018-0295"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2018-0295","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,12]]}}}