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Discrete Sobolev spaces and regularity of elliptic difference schemes

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Discrete Sobolev spaces and regularity of elliptic difference schemes RAIRO MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE ROB STEVENSON Discrete Sobolev spaces and regularity of elliptic difference schemes RAIRO – Modélisation mathématique et analyse numérique, tome 25, no 5 (1991), p. 607-640. <http://www.numdam.org/item?id=M2AN_1991__25_5_607_0> © AFCET, 1991, tous droits réservés. L’accès aux archives de la revue « RAIRO – Modélisation mathématique et analyse numérique » implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ rVfQfT] MATHEMATICAL MODEWNG AND NUMERICALANALYS1S ; f. i.L;\} j MODÉUSATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE (Vol. 25, n 5, 1991, p. 607 à 640) DISCRETE SOBOLEV SPACES AND REGULARITY OF ELLIPTIC DIFFERENCE SCHEMES (*) Rob STEVENSON (*) Communicated by V. THOMÉE Abstract. — This paper is concerned with the regularity of elliptic finite différence schemes wit h respect to discrete (fractional order) Sobolev spaces. For schemes arisingfrom discrétisations that are from the same « type » at the boundary as in the interior, it proves the discrete equivalent of Necas' regularity theoremfor differential operators on Lipschitz régions. A differentproof was given by Hackbusch. However, the proof her e is shorter and more transparent. In case of a curved boundary, usually different discrétisations are applied in points near the boundary. For schemes of this kind, it is shown by using Necas ' theorem for the corresponding « unperturbed » scheme, thaï «minimal» regularity implies the stronger regularity from Necas' theorem. Finally, conditions suffïcient for minimal regularity are given. Résumé. — Dans c

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