On the convergence of rational approximations of semigroups on intermediate spaces

Kovács, Mihály

doi:10.1090/S0025-5718-06-01905-3

On the convergence of rational approximations of semigroups on intermediate spaces

Author: Mihály Kovács
Journal: Math. Comp. 76 (2007), 273-286
MSC (2000): Primary 65J10; Secondary 65M12, 46N40, 46B70
DOI: https://doi.org/10.1090/S0025-5718-06-01905-3
Published electronically: October 4, 2006
MathSciNet review: 2261021
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Abstract: We generalize a result by Brenner and Thomée on the rate of convergence of rational approximation schemes for semigroups. Using abstract interpolation techniques we obtain convergence on a continuum of intermediate spaces between the Banach space and the domain of a certain power of the generator of the semigroup. The sharpness of the results is also discussed.

1. Nikolai Bakaev and Alexander Ostermann, Long-term stability of variable stepsize approximations of semigroups, Math. Comp. 71 (2002), no. 240, 1545–1567. MR 1933044, https://doi.org/10.1090/S0025-5718-01-01389-8
2. Philip Brenner and Vidar Thomée, Stability and convergence rates in 𝐿_{𝑝} for certain difference schemes, Math. Scand. 27 (1970), 5–23. MR 278549, https://doi.org/10.7146/math.scand.a-10983
3. Philip Brenner and Vidar Thomée, On rational approximations of semigroups, SIAM J. Numer. Anal. 16 (1979), no. 4, 683–694. MR 537280, https://doi.org/10.1137/0716051
4. Philip Brenner, Vidar Thomée, and Lars B. Wahlbin, Besov spaces and applications to difference methods for initial value problems, Lecture Notes in Mathematics, Vol. 434, Springer-Verlag, Berlin-New York, 1975. MR 0461121
5. Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York Inc., New York, 1967. MR 0230022
6. Paul L. Butzer and Rolf J. Nessel, Fourier analysis and approximation, Academic Press, New York-London, 1971. Volume 1: One-dimensional theory; Pure and Applied Mathematics, Vol. 40. MR 0510857
7. M. Crouzeix, S. Larsson, S. Piskarëv, and V. Thomée, The stability of rational approximations of analytic semigroups, BIT 33 (1993), no. 1, 74–84. MR 1326004, https://doi.org/10.1007/BF01990345
8. Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
9. Simone Flory, Frank Neubrander, and Lutz Weis, Consistency and stabilization of rational approximation schemes for 𝐶₀-semigroups, Evolution equations: applications to physics, industry, life sciences and economics (Levico Terme, 2000) Progr. Nonlinear Differential Equations Appl., vol. 55, Birkhäuser, Basel, 2003, pp. 181–193. MR 2013190
10. E. Hausenblas, A note on space approximation of parabolic evolution equations, Appl. Math. Comput. 157 (2004), no. 2, 381–392. MR 2088261, https://doi.org/10.1016/j.amc.2003.08.094
11. Reuben Hersh and Tosio Kato, High-accuracy stable difference schemes for well-posed initial value problems, SIAM J. Numer. Anal. 16 (1979), no. 4, 670–682. MR 537279, https://doi.org/10.1137/0716050
12. Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373
13. M. Kovács, A remark on the norm of integer order Favard spaces, Semigroup Forum, 71 (2005), 462-470.
14. M. Kovács, On positivity, shape, and norm-bound preservation of time-stepping methods for semigroups, J. Math. Anal. Appl. 304 (2005), 115-136.
15. M. Kovács, On qualitative properties and convergence of time-discretization methods for semigroups, Dissertation, Louisiana State University, 2004. For an electronic version, see http://etd.lsu.edu/docs/available/etd-07082004-143318/
16. M. Kovács and F. Neubrander, On the inverse Laplace-Stieltjes transform of A-stable rational functions, in preparation.
17. Peter D. Lax, Functional analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2002. MR 1892228
18. P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math. 9 (1956), 267–293. MR 79204, https://doi.org/10.1002/cpa.3160090206
19. A. Lunardi, Interpolation theory, Appunti, Scuola Normale Superiore, Pisa 1999. See also under http://prmat.math.unipr.it/ $\sim$ lunardi/LectureNotes/.
20. Rainer Nagel, Gregor Nickel, and Silvia Romanelli, Identification of extrapolation spaces for unbounded operators, Quaestiones Math. 19 (1996), no. 1-2, 83–100. MR 1390474
21. Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
22. Vidar Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR 1479170
23. David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923