×

On well-posedness of difference schemes for abstract parabolic equations in \(L^p([0,T];E)\) spaces. (English) Zbl 1022.65095

The paper deals with establishing of a coercive inequality in the discrete space \(L_{\tau_n}^p ([0,T];E_n)\). To this end the authors consider a general approximation scheme based on semigroup theory and functional analysis tools.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K90 Abstract parabolic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Amann H., Linear and Quasilinear Parabolic Problems (1995)
[2] Ashyralyev A., Ashgabat pp 79– (1993)
[3] Ashyralyev A., IZV. Akad. Nauk Turkmen. SSR Ser. Fiz.-Tekhn. Khim. Geol. Nauk 1 pp 3– (1989)
[4] Ashyralyev A., IZV. Akad. Nauk. Turkmen. SSR Ser. Fiz.-Tekhn. Khim. Geol. Nauk 3 pp 3– (1989)
[5] Ashyralyev A., Ukraine Math J. 44 pp 1466– (1992)
[6] DOI: 10.1155/S1085337501000495 · Zbl 0996.35027
[7] Ashyralyev A., Izv. Akad. Nauk Turkmen. SSR Ser. Fiz.-Tekhn. Khim. Geol. Nauk 6 pp 10– (1981)
[8] Ashyralyev A., Dokl. Akad. Nauk SSSR 275 pp 1289– (1984)
[9] Ashyralyev A., Izv. Akad. Nauk Turkmen. SSR Ser. Fiz.-Tekhn. Khim. Geol. Nauk 6 pp 3– (1985)
[10] Ashyralyev A., Dokl. Akad. Nauk Ukrainian SSR, Ser. Fiz.-Math. and Tech. Sciences 6 pp 3– (1988)
[11] DOI: 10.1016/0362-546X(94)E0004-Z · Zbl 0818.65047
[12] Ashyralyev A., Advances and Applications (1994)
[13] DOI: 10.4064/sm146-2-3 · Zbl 0981.39009
[14] DOI: 10.4064/sm146-1-3 · Zbl 1011.47030
[15] DOI: 10.1137/0716051 · Zbl 0413.41011
[16] DOI: 10.1007/BF01990345 · Zbl 0783.65050
[17] DOI: 10.1007/BF00046635 · Zbl 0802.47039
[18] DOI: 10.2969/jmsj/02840749 · Zbl 0331.65076
[19] Fujita H., Evolution Problems, Volume 2 of Handbook of Numerical Analysis, Chapter Finite Element Methods (part 1) (1991)
[20] Goldstein J.A., Oxford Mathematical Monographs (1985)
[21] DOI: 10.1007/BF01397374
[22] DOI: 10.1007/BF01437409
[23] DOI: 10.1007/BF01419530 · Zbl 0391.65022
[24] Grigorieff R.D., Numerical Treatment of Differential Equations (Halle, 1989), volume 121 of Teubner-Texte Math. pp 204– (1991)
[25] Guidetti D., On Maximal Regularity of Difference Schemes for Parabolic Equations in Spaces. N 8 · Zbl 1205.35156
[26] Guidetti D., Proccedings of International Conference Function Spaces. Differential Operators. Problems in Mathematical Education 2 pp 2231– (1998)
[27] Guidetti D., Progress in Partial Differential Equations (Pont-à-Mousson, 1997) 1 pp 167– (1998)
[28] DOI: 10.1137/0716050 · Zbl 0419.65036
[29] DOI: 10.1007/PL00004816
[30] Kalton N., The Calculus and Sums of Closed Operators · Zbl 0992.47005
[31] Kato T., Perturbation Theory for Linear Operators, Classics in Mathematics (1995) · Zbl 0836.47009
[32] Krein S.G., Linear Differential Equations in Banach Space (1971)
[33] Kunstmann P., Perturbation Theorems for Maximal Regularity · Zbl 1065.47008
[34] DOI: 10.1007/PL00004688 · Zbl 0927.47010
[35] Lunardi A., Analytic Semigroups and Optimal Regularity in Parabolic Problems, volume 16 of Progress in Nonlinear Differential Equations and Their Applications (1995) · Zbl 0816.35001
[36] Lyubich Y., Studia Math. 134 pp 153– (1999)
[37] DOI: 10.1007/978-3-0348-8547-8
[38] Nevanlinna O., Linear Operators(Warsaw, 1994) pp 247– (1997)
[39] DOI: 10.4064/sm145-2-2 · Zbl 0981.47002
[40] DOI: 10.1137/0730071 · Zbl 0794.65053
[41] Piskarev S., Tartu Riikl. Ül. Toimetised 492 pp 3– (1979)
[42] Piskarev S., Izv. Vyssh. Uchebn. Zaved. Mat. 4 pp 33– (1979)
[43] Piskarev S., Differentsial’ nye Uravneniya 20 pp 689– (1984)
[44] Piskarev, S. 1986.Principles of Discretization Methods III. Report ak–341087USSR: Acoustic Institute, Academy of Scince.
[45] Piskarev S., Differentsial’ nye Uravneniya 27 pp 1245– (1991)
[46] Polichka A.E., Trudy Mosk. Mat. Obshch. 36 pp 29– (1978)
[47] DOI: 10.1007/BF01399420 · Zbl 0307.65025
[48] DOI: 10.1007/BF01419524 · Zbl 0322.41010
[49] Samarskii A., Méthodes aux Différences pour Équations Elliptiques (1978)
[50] Samarskii A., Stability of Difference Schemes (1973) · Zbl 0368.65031
[51] Sobolevskii P.E., Dokl. Akad. Nauk SSSR 201 pp 1063– (1971)
[52] Sobolevskii P.E., Variational-Difference Methods in Mathematical Physics, Vychisl. Tsentr Sibirsk (1973)
[53] Sobolevskii P.E., Ukrain. Mat. Zh. 33 pp 39– (1981) · Zbl 0484.65033
[54] Strkalj Z., On Operator-Valued Fourier Multiplier Theorems · Zbl 1209.42005
[55] Stummel F., Linear Operators and Approximation(Proc. Conf. Oberwolfach, 1971), Internat. Ser. Numer. Math. 20 pp 196– (1972)
[56] Twizell E.H., Computational Methods for Partial Differential Equations (1984) · Zbl 0546.65062
[57] Ushijima T., Japan. J. Math. (N.S.) 1 pp 185– (1975)
[58] Vainikko G., Funktionalanalysis der Diskretisierungsmethoden (1976) · Zbl 0343.65023
[59] DOI: 10.1016/0362-546X(78)90013-5 · Zbl 0401.65034
[60] Vainikko G., The Use of Finite Element Method and Finite Difference Method in Geophysics (Proc. Summer School, Liblice, 1977) pp 173– (1978)
[61] Vasilev V.V., Mathematical Analysis 28 · Zbl 0442.93020
[62] Weis L., Proc of the 6th Internat. Conf on Evolution Equations (2000)
[63] DOI: 10.1007/PL00004457 · Zbl 0989.47025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.