{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T12:41:07Z","timestamp":1774528867295,"version":"3.50.1"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2014,4,1]]},"abstract":"<jats:title>Abstract.<\/jats:title>\n               <jats:p>\nThis paper is concerned with the stability of Petrov\u2013Galerkin discretizations with application\nto parabolic evolution problems in space-time weak form.\nWe will prove that the discrete inf-sup condition for an a priori fixed Petrov\u2013Galerkin discretization\nis satisfied uniformly under standard approximation and smoothness\nconditions without any further coupling between the discrete trial and test spaces\nfor sufficiently regular operators.\nIt turns out that one needs to choose different discretization levels for the trial and test spaces\nin order to obtain a positive lower bound for the discrete inf-sup condition\nwhich is independent of the discretization levels.\nIn particular, we state the required number of extra layers in order to guarantee uniform boundedness of the discrete inf-sup constants explicitly. This general result will be applied to\nthe space-time weak formulation of parabolic evolution problems as an important model example.\nIn this regard, we consider suitable hierarchical\nfamilies of discrete spaces.\nThe results apply, e.g., for finite element discretizations as well as for wavelet discretizations.\nDue to the Riesz basis property, wavelet discretizations allow for optimal\npreconditioning independently of the grid spacing.\nMoreover, our predictions on the stability, especially in view of\nthe dependence on the refinement levels w.r.t. the test and trial\nspaces, are underlined by numerical results.\nFurthermore, it can be observed that choosing the same discretization levels\nwould, indeed, lead to stability problems.\n<\/jats:p>","DOI":"10.1515\/cmam-2014-0001","type":"journal-article","created":{"date-parts":[[2014,1,20]],"date-time":"2014-01-20T20:07:02Z","timestamp":1390248422000},"page":"231-255","source":"Crossref","is-referenced-by-count":17,"title":["Stability of Petrov\u2013Galerkin Discretizations: Application to the Space-Time Weak Formulation for Parabolic Evolution Problems"],"prefix":"10.1515","volume":"14","author":[{"given":"Christian","family":"Mollet","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Mathematik, Universit\u00e4t Paderborn, Warburger Str. 100, 33098 Paderborn, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2014,1,16]]},"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/14\/2\/article-p231.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2014-0001\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2014-0001\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T21:32:28Z","timestamp":1680298348000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2014-0001\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,1,16]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2014,3,13]]},"published-print":{"date-parts":[[2014,4,1]]}},"alternative-id":["10.1515\/cmam-2014-0001"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2014-0001","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,1,16]]}}}