Stability of rational multistep approximations of holomorphic semigroups

Palencia, C.

doi:10.1090/S0025-5718-1995-1277770-2

Stability of rational multistep approximations of holomorphic semigroups
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Math. Comp. 64 (1995), 591-599 Request permission

Abstract:

In this paper we prove the stability of semidiscretizations in time of holomorphic semigroups in Banach spaces by means of ${\text {A}}(\alpha )$-stable rational multistep methods. No assumptions on the method other than ${\text {A}}(\alpha )$-stability are required. Our result is applicable in the maximum norm analysis of parabolic problems.

References

Philip Brenner and Vidar Thomée, On rational approximations of semigroups, SIAM J. Numer. Anal. 16 (1979), no. 4, 683–694. MR 537280, DOI 10.1137/0716051
Michel Crouzeix, On multistep approximation of semigroups in Banach spaces, Proceedings of the 2nd international conference on computational and applied mathematics (Leuven, 1986), 1987, pp. 25–35. MR 920377, DOI 10.1016/0377-0427(87)90123-3
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, DOI 10.1007/BF01990345

Approximation d’équations d’évolution linéaires par des méthodes multipas

Germund Dahlquist, Huang Mingyou, and Randall LeVeque, On the uniform power-boundedness of a family of matrices and the applications to one-leg and linear multistep methods, Numer. Math. 42 (1983), no. 1, 1–13. MR 716470, DOI 10.1007/BF01400914
Hector O. Fattorini, The Cauchy problem, Encyclopedia of Mathematics and its Applications, vol. 18, Addison-Wesley Publishing Co., Reading, Mass., 1983. With a foreword by Felix E. Browder. MR 692768
E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1987. Nonstiff problems. MR 868663, DOI 10.1007/978-3-662-12607-3
E. Hairer and G. Wanner, Solving ordinary differential equations. II, Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1991. Stiff and differential-algebraic problems. MR 1111480, DOI 10.1007/978-3-662-09947-6
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
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, DOI 10.1137/0716050
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Stig Larsson, Vidar Thomée, and Lars B. Wahlbin, Finite-element methods for a strongly damped wave equation, IMA J. Numer. Anal. 11 (1991), no. 1, 115–142. MR 1089551, DOI 10.1093/imanum/11.1.115
Marie-Noëlle Le Roux, Semidiscretization in time for parabolic problems, Math. Comp. 33 (1979), no. 147, 919–931. MR 528047, DOI 10.1090/S0025-5718-1979-0528047-2
Christian Lubich and Olavi Nevanlinna, On resolvent conditions and stability estimates, BIT 31 (1991), no. 2, 293–313. MR 1112225, DOI 10.1007/BF01931289
C. Palencia, A stability result for sectorial operators in Banach spaces, SIAM J. Numer. Anal. 30 (1993), no. 5, 1373–1384. MR 1239826, DOI 10.1137/0730071
C. Palencia, On the stability of variable stepsize rational approximations of holomorphic semigroups, Math. Comp. 62 (1994), no. 205, 93–103. MR 1201070, DOI 10.1090/S0025-5718-1994-1201070-9
C. Palencia and B. García-Archilla, Stability of linear multistep methods for sectorial operators in Banach spaces, Appl. Numer. Math. 12 (1993), no. 6, 503–520. MR 1232713, DOI 10.1016/0168-9274(93)90067-2

Multistep method and parabolic equations

Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
Anthony F. Ruston, Fredholm theory in Banach spaces, Cambridge Tracts in Mathematics, vol. 86, Cambridge University Press, Cambridge, 1986. MR 861275, DOI 10.1017/CBO9780511569180
J. M. Sanz-Serna and C. Palencia, A general equivalence theorem in the theory of discretization methods, Math. Comp. 45 (1985), no. 171, 143–152. MR 790648, DOI 10.1090/S0025-5718-1985-0790648-7