On well-posedness of difference schemes for abstract parabolic equations in Lp([0, T];E) spaces

Numerical Functional Analysis and Optimization

Volumn 23, Issue 7-8, 2002, Pages 669-693

  Ashyralyev, Allaberen ^a   Piskarev, Serguei ^b   Weis, Lutz ^c  

^a FATIH UNIVERSITY   (Turkey)

^b MOSCOW STATE UNIVERSITY   (Russian Federation)

^c UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

Abstract differential equations; Banach spaces; C0 semigroups; Coercivity; Difference schemes; Discrete semigroups; Maximal regularity inequality; Mikhlin's multiplier theorem; Rational approximations; Semidiscretization; UMD spaces

Indexed keywords

APPROXIMATION THEORY; BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; EXTRAPOLATION; FUNCTION EVALUATION; INTERPOLATION; PARTIAL DIFFERENTIAL EQUATIONS; SET THEORY; THEOREM PROVING;

ABSTRACT PARABOLIC DIFFERENTIAL EQUATIONS; BANACH SPACES; MAXIMAL REGULARITY INEQUALITY; MIKHLIN MULTIPLIER THEOREM; RATIONAL APPROXIMATIONS; SEMIDISCRETIZATION; SEMIGROUP THEORY; WELL POSEDNESS;

FINITE DIFFERENCE METHOD;

EID: 0036868462     PISSN: 01630563     EISSN: None     Source Type: Journal    
DOI: 10.1081/NFA-120016264     Document Type: Article

References (63)

1. Amann, H. Linear and Quasilinear Parabolic Problems, Vol. 1, Birkhäuser Verlag: Basel, 1995; Abstract Linear Theory.
2. Ashyralyev, A. Single-Step Difference Schemes Generated by Taylors Decomposition in the Two Points. Proceedings of the Scientific-Practical Conference on Differential Equation and Their Applications. Ashgabat 1993, 79-81 (Russian).
3. Ashyralyev, A. Estimations of the Convergence for the Solution of the Modification Crank-Nicolson Difference Schemes for the Parabolic Equations with the Nonsmooth Initial Data. IZV. Akad. Nauk Turkmen. SSR Ser. Fiz.-Tekhn. Khim. Geol. Nauk 1989, 1, 3-8 (Russian).
4. Ashyralyev, A. Coercive Solvability of the Parabolic Equations in the Spaces of the Smooth Functions. IZV. Akad. Nauk. Turkmen. SSR Ser. Fiz.-Tekhn. Khim. Geol. Nauk 1989, 3, 3-13 (Russian).
5. Ashyralyev, A. Correct Solvability of the Pade Difference Schemes for the Parabolic Equations in Holder Spaces. Ukraine Math J. 1992, 44(11), 1466-1476 (Russian).
6. Ashyralyev, A.; Hanalyev, A.; Sobolevskii, P.E. Coercive Solvability of Nonlocal Boundary Value Problem for Parabolic Equations. Abstract and Applied Analysis 2001, 6(1), 53-61.
7. Ashyralyev, A. Sobolevskii, P.E. Correct Solvability of the Crank-Nicolson Difference Scheme for the Parabolic Equations. Izv. Akad. Nauk Turkmen. SSR Ser. Fiz.-Tekhn. Khim. Geol. Nauk 1981, 6, 10-16 (Russian).
8. Ashyralyev, A.; Sobolevskii, P.E. Theory of the Interpolation of the Linear Operators and the Stability of the Difference Schemes. Dokl. Akad. Nauk SSSR 1984, 275(6), 1289-1291 (Russian).
9. Ashyralyev, A.; Sobolevskii, P.E. Coercive Stability of the Difference Schemes of the First and Second Order of the Approximation with the Coefficients Independent of the Time, Izv. Akad. Nauk Turkmen. SSR Ser. Fiz.-Tekhn. Khim. Geol. Nauk 1985, 6, 3-11 (Russian).
10. Ashralyev, A.; Sobolevskii, P.E. Difference Schemes of a High Order of Accuracy for the Parabolic Equations with the Variable Coefficients. Dokl. Akad. Nauk Ukrainian SSR, Ser. Fiz.-Math. and Tech. Sciences 1988, 6, 3-7 (Russian).
11. Ashyralyev, A.; Sobolevskii, P.E. Well-Posed Solvability of the Cauchy Problem for the Difference Equations of the Parabolic Type. Nonlinear Analysis. Theory Methods and Applications 1995, 24(2), 257-264.
12. Ashyralyev, A.; Sobolevskiǐ, P.E. Well-Posedness of Parabolic Difference Equations, volume 69 of Operator Theory: Advances and Applications; Birkhäuser Verlag: Basel, 1994; Translated from the Russian by A. Iacob.
13. Blunck, S. Maximal Regularity of Discrete and Continuous Time Evolution Equations. Studia Math. 2001, 146(2), 157-176.
14. Blunck, S.; Weis, L. Operator Theoretic Properties of Semigroups in Terms of Their Generators. Studia Math. 2001, 146(1), 35-54.
15. Brenner, P.; Thomée, V. On Rational Approximations of Semigroups. SIAM J. Numer. Anal. 1979, 16, 683-694.
16. Crouziex, M.; Larsson, S.; Piskarev, S.; Thomée, V. The Stability of Rational Approximations of Analytic Semigroups. BIT 1993, 33(1), 74-84.
17. Eberhardt, B.; Greiner, G. Baillon's Theorem on Maximal Regularity. Acta Appl. Math. 1992, 27, 47-54. Positive Operators and Semigroups on Banach Lattices (Curaçao, 1990).
18. Fujita, H.; Mizutani, A. On the Finite Element Method for Parabolic Equations, I; Approximation of Holomorphic Semi-groups. J. Math. Soc. Japan 1976, 28(4), 749-771.
19. Fujita, H.; Suzuki, I. Evolution Problems, Volume 2 of Handbook of Numerical Analysis, Chapter Finite Element Methods (part 1); Elsevier Science Publishers B.V.: North Holland, 1991.
20. Goldstein, J.A. Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs; The Clarendon Press Oxford University Press: New York, 1985.
21. Grigorieff, R.D. Diskrete Approximation von Eigenwertproblemen. I. Qualitative Konvergenz. Numer. Math. 1975, 24(4), 355-374.
22. Grigorieff, R.D. Diskrete Approximation von Eigenwertproblemen. II. Konvergenzordnung. Numer. Math. 1975, 24(5), 415-433.
23. Grigorieff, R.D. Diskrete Appoximation von Eigenwertproblemen. III. Asymptotische Entwicklungen. Numer. Math. 1975/76, 25(1), 79-97.
24. Grigorieff, R.D. Time Discretization of Semigroups by the Variable Two-Step BDF Method. In Numerical Treatment of Differential Equations (Halle, 1989), volume 121 of Teubner-Texte Math.; Teubner: Stuttgart, 1991; 204-216.
25. θ (Ω) Spaces. N 8, Dipartimento di Matematica, Universitaá degli Studi di Bologna, 1997. Preprint.
26. θ (Ω) Space. In Proccedings of International Conference Function Spaces. Differential Operators. Problems in Mathematical Education RUDN, Moscow, 1998, Vol. 2, 2231-2237.
27. θ (Ω). In Progress in Partial Differential Equations, (Pont-à-Mousson, 1997); Longman: Harlow, 1998; Vol. 1, 167-180.
28. Hersh, R.; Kato, T. High-Accuracy Stable Difference Schemes for Well-Posed Initial Value Problems. SIAM J. Numer. Anal. 1979, 16(4), 670-682.
29. p-Maximal Regularity. Math. Z. 2000, 235(3), 559-568.
30. ∞ Calculus and Sums of Closed Operators. Submitted.
31. Kato, T. Perturbation Theory for Linear Operators, Classics in Mathematics; Springer-Verlag: Berlin, 1995. Reprint of the 1980 edition.
32. Krein, S.G. Linear Differential Equations in Banach Space; American Mathematical Society: Providence, R.I., 1971. Translated from the Russian by Danskin, J.M., Translations of Mathematical Monographs, Vol. 29.
33. Kunstmann, P.; Weiss, L. Perturbation Theorems for Maximal Regularity. Submitted
34. Le Merdy, C. Counterexamples on Lp-Maximal Regularity. Math. Z. 1999, 230, 47-62.
35. Lunardi, A. Analytic Semigroups and Optimal Regularity in Parabolic Problems, volume 16 of Progress in Nonlinear Differential Equations and Their Applications; Birkhäuser Verlag: Basel, 1995.
36. Lyubich, Y. Spectral Localization, Power Boundedness and Invariant Subspaces Under Ritt's Type Condition. Studia Math. 1999, 134(2), 153-167.
37. Nevanlinna, O. Convergence of Iterations for Linear Equations; Birkhäuser Verlag: Basel, 1993.
38. Nevanlinna, O. On the Growth of the Resolvent Operators for Power Bounded Operators. In Linear Operators (Warsaw, 1994); Polish Acad. Sci.: Warsaw, 1997; 247-264.
39. Nevanlinna, O. Resolvent Conditions and Powers of Operators. Studia Math. 2001, 145(2), 113-134.
40. Palencia, C. A Stability Result for Sectorial Operators in Banach Spaces. SIAM J. Numer. Anal. 1993, 30(5), 1373-1384.
41. Piskarev, S. Approximation of Holomorphic Semigroups. Tartu Riikl. Ül. Toimetised 1979, 492, 3-14.
42. Piskarev, S. Error Estimates in the Approximation of Semigroups of Operators by Padé Fractions. Izv. Vyssh. Uchebn. Zaved. Mat. 1979, 4, 33-38.
43. Piskarev, S. Stability of Difference Schemes in Cauchy Problems with Almost Periodic Solutions. Differentsial' nye Uravneniya 1984, 20(4), 689-695.
44. Piskarev, S. Principles of Discretization Methods III. Report ak-3410, Acoustic Institute, Academy of Science: USSR, 1986; 87 pages.
45. 0-Semigroups of Operators. Differentsial' nye Uravneniya 1991, 27(7), 1245-1250, 1287.
46. Polichka, A.E.; Sobolevskii, P.E. Correct Solvability of Parabolic Difference Equations in Bochner Spaces. Trudy Mosk. Mat. Obshch. 1978, 36, 29-57 (Russian).
47. -z by Rational Functions with Real Poles. Numer. Math. 1975/76, 25(3), 307-322.
48. z. Numer. Math. 1975/76, 25(1), 1-14.
49. Samarskii, A.; Andréev, V.; Méthodes aux Différences pour Équations Elliptiques; Éditions Mir: Moscow, 1978. Traduit du russe par Djilali Embarek.
50. Samarskii, A.; Gulin, A. V. Stability of Difference Schemes; Nauka: Moscow, 1973 (in Russian).
51. Sobolevskii, P.E. The Coercive Solvability of Difference Equations. Dokl. Akad. Nauk SSSR 1971, 201, 1063-1066.
52. Sobolevskii, P. E. On the Stability and Convergence of the Crank-Nicolson Scheme. In Variational-Difference Methods in Mathematical Physics, Vychisl. Tsentr Sibirsk; Otdel. Akad. Nauk SSSR: Novosibirsk, 1973 (1974) (Russian).
53. Sobolevskii, P.E.; Hoáng Vǎn Lai. Difference Schemes of Optimal Type for the Approximate Solution of Parabolic Equations (the Banach case). Ukrain. Mat. Zh. 1981, 33, 39-46.
54. Strkalj, Z.; Weis, L. On Operator-Valued Fourier Multiplier Theorems. Submitted.
55. Stummel, F. Diskrete Konvergenz Linearer Operatoren. III. In Linear Operators and Approximation (Proc. Conf. Oberwolfach, 1971), Internat. Ser. Numer. Math. Vol. 20.; Birkhäuser: Basel, 1972; 196-216.
56. Twizell, E.H. Computational Methods for Partial Differential Equations; Ellis Horwood Ltd.: Chichester, 1984.
57. Ushijima, T. Approximation Theory for Semi-groups of Linear Operators and Its Application to Approximation of Wave Equations. Japan. J. Math. (N.S.) 1975/76, 1(1), 185-224.
58. Vainikko, G. Funktionalanalysis der Diskretisierungsmethoden; B. G. Teubner Verlag: Leipzig, 1976. Mit Englischen und Russischen Zusammenfassungen, Teubner-Texte zur Mathematik.
59. Vainikko, G. Approximative Methods for Nonlinear Equations (Two Approaches to the Convergence Problem). Nonlinear Anal. 1978, 2, 647-687.
60. Vainikko, G. Foundations of Finite Difference Method for Eigenvalue Problems. In The Use of Finite Element Method and Finite Difference Method in Geophysics (Proc. Summer School, Liblice, 1977); Cesk. Akad. Věd; Prague, 1978; 173-192.
61. Vasilev, V.V.; Krein, S.G.; Piskarev, S. Operator Semigroups. Cosine Operator Functions, and Linear Differential Equations. In Mathematical Analysis, Vol. 28 (Russian), Itogi Nauki i Tekhniki, 87-202, 204; Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. Moscow, 1990, Translated in J. Soviet Math. 1991, 54(4), 1042-1129.
62. p Regularity. In Proc of the 6th Internat. Conf on Evolution Equations; Marcel Dekker, 2000.
63. p Regularity Mathematische Annalen 2001, 319, 735-758.