A parallel high-accuracy method for the first-order evolution equation in hilbert and banach spaces

Computational Methods in Applied Mathematics

Volumn 3, Issue 1, 2003, Pages 86-115

  Gavrilyuk, Ivan P ^a   Makarov, Vladimirl ^b   Ryabichev, Vyacheslav L ^c  

^a Staatliche Studienakademie Am Wartenberg 2   (Germany)

^b INSTITUTE OF MATHEMATICS   (Ukraine)

^c TARAS SHEVCHENKO NATIONAL UNIVERSITY OF KYIV   (Ukraine)

Author keywords

Error analysis; Evolution operator; First order evolution equation; Hilbert and Banach spaces; Parallel high accuracy method; Stability estimates; Strongly P positive operator

Indexed keywords

EID: 8644276122     PISSN: 16094840     EISSN: 16099389     Source Type: Journal    
DOI: 10.2478/cmam-2003-0008     Document Type: Article

References (19)

1. A. Ashyralyev and P. Sobolevskii, Well-Posedness of Parabolic Difference Equations, Birkhauser, 1994.
2. H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. 1, Mc Graw-Hill Book Company, Inc., 1953.
3. R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Evolutions problems I, Springer Verlag, 1992.
4. H. Fujita, N. Saito and T. Suzuki, Operator Theory and Numerical Methods, Elsevier, 2001.
5. I. P. Gavrilyuk, Strongly P-positive operators and explicit representation of the solutions of initial value problems for second order differential equations in Banach space. J. Math. Anal. Appl., 236 (1999), pp. 327-349.
6. I. P. Gavrilyuk and V. L. Makarov, Exponentially convergent parallel discretization methods for the first order evolution equations, Comput. Methods Appl. Math., 1 (2001), No. 4, pp. 333-355.
7. I. P. Gavrilyuk and V. L. Makarov, Explicit and approximate solutions of second order elliptic differential equations in Hilbert- and Banach spaces. Numer. Funct. Anal. Optimization, 20 (1999), pp. 695-717.
8. I. P. Gavrilyuk and V. L. Makarov, Exact and approximate solutions of some operator equations based on the Cayley transform. Linear Algebra Appl., 282 (1998), pp. 97-121.
9. I. P. Gavrilyuk and V. L. Makarov, Representation and approximation of the solution of an initial value problem for a first order differential eqation in Banach space. Z. Anal. Anwend., 15 (1996), pp. 495-527.
10. D. Gonzalez C. and Palencia C., Stability of time-stepping methods for abstract time-dependent parabolic problems, SIAM J. Numer. Anal., 35 (1998), No. 3, pp. 973-989.
11. D. Gonzalez C. and Palencia C., Stability of Runge-Kutta methods for abstract time-dependent parabolic problems, the Holder case, Math. Comput., 68 (1999), No. 225, pp. 73-89.
12. I. Ixaru, Numerical Methods for Differential Equations and Applications, D. Reidel Publishing Company, A member of the Kluwer Academic Publishers Group, Dordrecht/ Boston/ Lancaster, 1984.
13. Krein, S.G., Linear Differential Operators in Banach Spaces (Transl. Amer. Math. Soc., vol. 29), New York, Amer. Math. Soc., 1971.
14. Krein, S. G. et al. eds., Functional Analysis, Moscow, Nauka, 1972 (in Russian).
15. A. Pazy, Semigroups of Linear Operator and Applications to Partial Differential Equations, Springer Verlag, 1983.
16. A.A. Samarskii, I. P. Gavrilyuk and V. L. Makarov, Stability and regularization of three-level difference schemes with unbounded operator coefficients in Banach spaces, SIAM J. Numer. Anal., 39 (2001), No. 2, pp. 708-723.
17. A.A. Samarskii, P.N.Vabischevich and P. P. Matus, Difference Schemes with Operator Multipliers, ISBN 985-6366-13-5, Minsk, 1998 (in Russian).
18. D. Sheen, I. H. Sloan and V. Thomee, A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature, Math. Comput., 69 (2000), No. 229, pp. 177195.
19. F. Stenger, Numerical Methods Based on Sinc and Analytic Functions. Springer Verlag, 1993.