Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Published electronically: October 4, 2006
MathSciNet review: 2261021
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $ X$ and the domain of a certain power of the generator of the semigroup. The sharpness of the results is also discussed.

References [Enhancements On Off] (What's this?)

  • 1. N. Yu. Bakaev and A. Ostermann, Long-term stability of variable-stepsize approximation of semigroups, Math. Comp. 71 (2002), 1545-1567. MR 1933044 (2003j:65055)
  • 2. P. Brenner and V. Thomée, Stability and convergence rates in $ L_p$ for certain difference schemes, Math. Scand. 27 (1970), 5-23.MR 0278549 (43:4279)
  • 3. P. Brenner and V. Thomée, On rational approximation of semigroups, SIAM J. Numer. Anal. 16 (1979), 683-694. MR 0537280 (80j:47052)
  • 4. P. Brenner, V. Thomée and L. B. Wahlbin, Besov Spaces and Applications to Difference Methods for Initial Value Problems, Lecture Notes in Mathematics 434, Springer, 1975. MR 0461121 (57:1106)
  • 5. P. L. Butzer and H. Berens, Semi-groups of Operators and Approximation, Springer, 1967. MR 0230022 (37:5588)
  • 6. P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Academic Press, 1971. MR 0510857 (58:23312)
  • 7. M. Crouzeix, S. Larsson, S. Piskarev and V. Thomée, The stability of rational approximations of analytic semigroups, BIT 33 (1993), 74-84. MR 1326004 (96f:65069)
  • 8. K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, 2000. MR 1721989 (2000i:47075)
  • 9. S. Flory, F. Neubrander and L. Weis, Consistency and stabilization of rational approximation schemes for $ C_0$-semigroups, Progress in Nonlinear Differential Equations 55 (2003), 181-193. MR 2013190 (2005c:47056)
  • 10. E. Hausenblas, A note on space approximation of parabolic evolution equations, Appl. Math. Comp. 157 (2004), 381-392. MR 2088261 (2005g:34128)
  • 11. R. Hersh and T. Kato, High-accuracy stable difference schemes for well posed initial-value problems, SIAM J. Numer. Anal. 16 (1979), 671-682. MR 0537279 (80h:65036)
  • 12. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Colloquium Publications 31, American Mathematical Society, 1957.MR 0089373 (19:664d)
  • 13. M. Kovács, A remark on the norm of integer order Favard spaces, Semigroup Forum, 71 (2005), 462-470.
  • 14. M. Kovács, On positivity, shape, and norm-bound preservation of time-stepping methods for semigroups, J. Math. Anal. Appl. 304 (2005), 115-136.
  • 15. M. Kovács, On qualitative properties and convergence of time-discretization methods for semigroups, Dissertation, Louisiana State University, 2004. For an electronic version, see
  • 16. M. Kovács and F. Neubrander, On the inverse Laplace-Stieltjes transform of A-stable rational functions, in preparation.
  • 17. P. D. Lax, Functional Analysis, Wiley, 2002. MR 1892228 (2003a:47001)
  • 18. P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math. IX (1956), 267-293.MR 0079204 (18:48c)
  • 19. A. Lunardi, Interpolation theory, Appunti, Scuola Normale Superiore, Pisa 1999. See also under$ \sim$lunardi/LectureNotes/.
  • 20. R. Nagel, G. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaestiones Mathematicae 19 (1996), 83-100.MR 1390474 (97e:46102)
  • 21. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems, Wiley, 1967. MR 0220455 (36:3515)
  • 22. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, 1997.MR 1479170 (98m:65007)
  • 23. D. V. Widder, The Laplace Transform, Princeton University Press, 1946. MR 0005923 (3:232d)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65J10, 65M12, 46N40, 46B70

Retrieve articles in all journals with MSC (2000): 65J10, 65M12, 46N40, 46B70

Additional Information

Mihály Kovács
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803 and Department of Analysis, Mathematics Institute, University of Miskolc, Miskolc-Egyetemváros, Hungary, H-3515
Address at time of publication: Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand

Keywords: Rational approximation of semigroups, intermediate spaces, Favard spaces, Hille-Phillips functional calculus, time-discretization
Received by editor(s): September 7, 2005
Published electronically: October 4, 2006
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society