Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces

Mathematics of Computation

Volumn 74, Issue 250, 2005, Pages 555-583

  Gavrilyuk, Ivan P ^a   Makarov, Volodymyr L ^b  

^a Berufsakademie ThüRingen ^*  (Germany)

^b INSTITUTE OF MATHEMATICS   (Ukraine)

Author keywords

Algorithms without accuracy saturation; Evolution equation; Exponentially convergent algorithms; Parameter dependent operator; Sinc methods

Indexed keywords

EID: 14944374194     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-04-01720-X     Document Type: Article

References (30)

1. K.I. Babenko: Fundamentals of Numerical Analysis, Nauka, Moscow, 1986 (in Russian). MR0889669 (88g:65001)
2. H. Bateman and A. Erdélyi: Higher transcendental functions, vol. 2, McGraw-Hill Book Company, Inc., 1953. MR0058756 (15:419i)
3. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang: Spectral Methods in Fluid Dynamics, Springer-Verlag 1988. MR0917480 (89m:76004)
4. P.G. Ciarlet and J.-L. Lions (eds.): Handbook of Numerical Analysis, Vol. 1, North-Holland, Amsterdam, 1990. MR1039323 (91f:65001)
5. R. Dautray and J.-L. Lions: Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Evolution Problems I, Springer-Verlag 1992. MR1156075 (92k:00006)
6. H. Fattorini: Second Order Linear Differential Equations in Banach Spaces, North-Holland, 1985, pp. 165-237. MR0797071 (87b:34001)
7. H. Fujita, N. Saito and T. Suzuki: Operator Theory and Numerical Methods, Elsevier, 2001. MR1854280 (2002i:65111)
8. 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. Journ. of Math. Analysis and Appl. 236 (1999), 327-349. MR1704587 (2001j:34072)
9. I.P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij: ℋ-Matrix Approximation for the Operator Exponential with Applications. Numer. Math. 92 (2002) 1, 83-111. MR1917366 (2003g:65061)
10. I.P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij: ℋ-Matrix approximation for elliptic solution operators in cylinder domains. East-West J. of Num. Math., vol. 9(2001), No. 1, pp. 25-59. MR1839197 (2002e:65064)
11. I.P. Gavrilyuk, W. Hackbusch, and B. Khoromskij: Data-Sparse Approximation to a Class of Operator-Valued Functions, Math. Comp., posted on August 23, 2004, PII S00255718(04)01703-X (to appear in print).
12. I.P. Gavrilyuk, W. Hackbusch, B. Khoromskij: Data-Sparse Approximation to the Operator-Valued Functions of Elliptic Operators, Math. Comp,. 73 (2004) 1297-1324.
13. I.P. Gavrilyuk and V.L. Makarov: Exponentially convergent parallel discretization methods for the first order evolution equations, Computational Methods in Applied Mathematics, Vol. 1, No. 4, 2001, pp. 333-355. MR1892950 (2003f:65174)
14. 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), 495-527. MR1394440 (97h:65076)
15. I.P. Gavrilyuk, V.L. Makarov and V.L. Rjabichev: A parallel method of high accuracy for the first order evolution equation in Hilbert and Banach space. Computational Methods in Applied Mathematics, Vol. 3, No. 1, 2003, pp. 86-115. MR2002259 (2004g:65079)
16. W. Hackbusch: A sparse matrix arithmetic based on ℋ-matrices. Part I: Introduction to ℋ-matrices. Computing 62 (1999), 89-108. MR1694265 (2000c:65039)
17. W. Hackbusch and B. N. Khoromskij: A sparse ℋ-matrix arithmetic. Part II: Application to multi-dimensional problems. Computing 64 (2000), 21-47. MR1755846 (2001i:65053)
18. S.G. Krein: Linear Differential Operators in Banach Spaces, Moscow: Nauka, 1967 (in Russian). MR0247239 (40:508)
19. S.G. Krein: Linear Differential Operators in Banach Spaces (Transl. Amer. Math. Soc.: vol. 29). New York: Amer. Math. Soc., 1971. MR0342804 (49:7548)
20. I.P. Natanson: Constructive function theory, Moscow, 1949. MR0034464 (11:591c)
21. I.P. Natanson: Constructive function theory. V. 1: Uniform approximation, translated from Russian by N. Obolensky. V. 2: Approximation in mean, translated from Russian by John R. Schulenberger. V. 3: Interpolation and approximation quadratures, translated from Russian by John R. Schulenberger, Ungar, New York, 1964, 1965, 1965. MR0196340 (33:4529a); MR0196341 (33:4529b); MR0196342 (33:4529c)
22. I.P. Natanson: Konstruktive Funktionentheorie, Akademie-Verlag, Berlin, 1955. MR0069915 (16:1100d)
23. A. Pazy: Semigroups of linear operator and applications to partial differential equations, Springer Verlag, 1983. MR0710486 (85g:47061)
24. 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 Journal on Numerical Analysis, v. 39, No. 2, 2001, pp. 708-723. MR1860256 (2002h:35163)
25. A.A. Samarskii, P.N. Vabischevich and P.P. Matus: Difference Schemes with Operator Factors, Kluwer Academic Publishers, Boston/Dordrecht/London, 2002. MR1950844 (2003k:65095)
26. F. Stenger: Numerical methods based on Sinc and analytic functions. Springer Verlag, 1993. MR1226236 (94k:65003)
27. F. Stenger, A. Naghsh-Nilchi, J. Niebsch, and R. Ramlau: Sampling Methods for Approximate Solution of PDE, in "Inverse Problems and Image Analysis" (eds. Z. Nashed and O. Scherzer), AMS, 2002, pp. 199-250. MR1940998 (2003m:35013)
28. G. Szegö: Orthogonal Polynomials. American Mathematical Society, New York, 1959. MR0106295 (21:5029)
29. G. Szegö: Orthogonal Polynomials (with an Introduction and a Complement by J.L. Geronimus). State Publishing House of Physical and Mathematical Literature, Moscow, 1962.
30. J.A. Goldstein: Semigroups of Linear Operators and Applications, Oxford University Press, New York, and Clarendon Press, Oxford, 1985. MR0790497 (87c:47056)