{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T16:47:12Z","timestamp":1648658832023},"reference-count":26,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2011,2,14]],"date-time":"2011-02-14T00:00:00Z","timestamp":1297641600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Numerical Analysis"],"published-print":{"date-parts":[[2011,2,14]]},"abstract":"<jats:p>This paper is devoted to the numerical analysis of abstract parabolic problem ;\n, with hyperbolic generator . We are developing a general approach to establish\na discrete dichotomy in a very general setting in case of discrete approximation in space and\ntime. It is a well-known fact that the phase space in the neighborhood of the\nhyperbolic equilibrium can be split in a such way that the original initial value problem is reduced\nto initial value problems with exponential decaying solutions in opposite time direction. We use\nthe theory of compact approximation principle and collectively condensing approximation to\nshow that such a decomposition of the flow persists under rather general approximation schemes. \nThe main assumption of our results is naturally satisfied, in particular, for operators with\ncompact resolvents and condensing semigroups and can be verified for finite element as well as\nfinite difference methods.<\/jats:p>","DOI":"10.1155\/2011\/582740","type":"journal-article","created":{"date-parts":[[2012,2,14]],"date-time":"2012-02-14T21:02:11Z","timestamp":1329253331000},"page":"1-22","source":"Crossref","is-referenced-by-count":1,"title":["The Exponential Dichotomy under Discretization on General Approximation Scheme"],"prefix":"10.1155","volume":"2011","author":[{"given":"Javier","family":"Pastor","sequence":"first","affiliation":[{"name":"Department de Mathem\u00e0tica Aplicada, University de Val\u00e8ncia, 46100 Burjassot, Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergey","family":"Piskarev","sequence":"additional","affiliation":[{"name":"Scientific Research Computer Center, Moscow State University, Vorobjevy Gory, Moscow 119899, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"98","reference":[{"key":"19","series-title":"Lecture Notes in Mathematics","volume":"840","year":"1981"},{"key":"23","year":"1971"},{"key":"34","year":"2005"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1994.1109"},{"key":"45","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-07-08691-1"},{"key":"46","series-title":"Lecture Notes in Pure and Applied Mathematics","volume-title":"The spectral radius, hyperbolic operators and Lyapunov's theorem","volume":"215","year":"2001"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.3934\/dcdsb.2008.10.19"},{"key":"30","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1988-0958058-1"},{"issue":"2","key":"5","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1134\/S1064562407050080","volume":"76","year":"2007","journal-title":"Doklady Mathematics"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1134\/S0012266110020072"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1023\/A:1012832208161"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1007\/BF01437409"},{"key":"40","series-title":"International Series of Numerical Mathematics","volume-title":"Diskrete Konvergenz linearer operatoren. 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