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Upper semicontinuity of attractors for linear multistep methods approximating sectorial evolution equations

Authors: Adrian T. Hill and Endre Süli
Journal: Math. Comp. 64 (1995), 1097-1122
MSC: Primary 65J05; Secondary 34G20, 47H20, 47N20, 58F13, 65L06, 65M12
MathSciNet review: 1297470
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Abstract: This paper sets out a theoretical framework for approximating the attractor $ \mathcal{A}$ of a semigroup $ S(t)$ defined on a Banach space X by a q-step semidiscretization in time with constant steplength k. Using the one-step theory of Hale, Lin and Raugel, sufficient conditions are established for the existence of a family of attractors $ \{ {\mathcal{A}_k}\} \subset {X^q}$, for the discrete semigroups $ S_k^n$ defined by the numerical method. The convergence properties of this family are also considered. Full details of the theory are exemplified in the context of strictly $ A(\alpha )$-stable linear multistep approximations of an abstract dissipative sectorial evolution equation.

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  • [1] W.-J. Beyn, On invariant closed curves for one-step methods, Numer. Math. 51 (1987), 103-122. MR 884136 (88d:65121)
  • [2] J. C. Butcher, On the convergence of numerical solutions to ordinary differential equations, Math. Comp. 20 (1966), 1-10. MR 0189251 (32:6678)
  • [3] M. Crouzeix, On multistep approximation of semigroups in Banach space, J. Comput. Appl. Math. 20 (1987), 25-35. MR 920377 (88j:65135)
  • [4] J. Dieudonné, Treatise on analysis, Vol. III, Academic Press, New York, 1972. MR 0350769 (50:3261)
  • [5] N. Dunford and J. T. Schwartz, Linear operators Part I: General theory, Wiley-Interscience, New York, 1958. MR 1009162 (90g:47001a)
  • [6] T. Eirola and O. Nevanlinna, What do multistep methods approximate?, Numer. Math. 53 (1988), 559-569. MR 954770 (89i:65069)
  • [7] C. M. Elliott and A. M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal. 30 (1993), 1622-1663. MR 1249036 (94j:65127)
  • [8] A. Friedman, Partial differential equations, Holt, New York, 1969. MR 0445088 (56:3433)
  • [9] D. F. Griffiths and A. R. Mitchell, Stable periodic bifurcations of an explicit discretization of a nonlinear partial differential equation in reaction-diffusion, IMA J. Numer. Anal. 8 (1988), 435-454. MR 975605 (90h:65146)
  • [10] E. Hairer, A. Iserles, and J. M. Sanz-Serna, Equilibria of Runge-Kutta methods, Numer. Math. 58 (1990), 243-254. MR 1075156 (91i:65126)
  • [11] E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations, Vol. I, Springer, Berlin, 1987.
  • [12] E. Hairer and G. Wanner, Solving ordinary differential equations, Vol. II, Springer, Berlin, 1991. MR 1111480 (92a:65016)
  • [13] J. K. Hale, Asymptotic behaviour of dissipative systems, Math. Surveys Monographs, vol. 25, Amer. Math. Soc., Providence, RI, 1988. MR 941371 (89g:58059)
  • [14] J. K. Hale, X.-B. Lin, and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp. 50 (1988), 89-123. MR 917820 (89a:47093)
  • [15] J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. 154 (1989), 281-326. MR 1043076 (91f:58087)
  • [16] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, Springer, Berlin, 1981. MR 610244 (83j:35084)
  • [17] A. T. Hill, Attractors for nonlinear convection-diffusion equations and their numerical approximation, D. Phil. Thesis, Oxford, 1992.
  • [18] A. T. Hill and E. Süli, Upper semicontinuity of attractors for linear multistep methods, University of Bath Mathematics Preprint 94/11.
  • [19] E. Isaacson and H. B. Keller, Analysis of numerical methods, Wiley, New York, 1966. MR 0201039 (34:924)
  • [20] A. Iserles, Stability and dynamics for nonlinear ordinary differential equations, IMA J. Numer. Anal. 10 (1990), 1-30. MR 1036645 (91e:65105)
  • [21] A. Iserles, A. T. Peplow, and A. M. Stuart, A unified approach to spurious solutions introduced by time discretization. Part I: basic theory, SIAM J. Numer. Anal. 28 (1991), 1723-1751. MR 1135763 (92h:65102)
  • [22] A. Iserles and A. M. Stuart, Unified approach to spurious solutions introduced by time-discretization. Part II: BDF-like methods, IMA J. Numer. Anal. 12 (1992), 487-502. MR 1186731 (93i:65070)
  • [23] U. Kirchgraber, Multistep methods are essentially one-step methods, Numer. Math. 48 (1986), 85-90. MR 817122 (87d:65073)
  • [24] U. Kirchgraber, F. Lasagni, K. Nipp, and D. Stoffer, On the application of the invariant manifold theory, in particular to numerical analysis, Internat. Ser. Numer. Math. 97 (1991), 189-197. MR 1109521 (92h:58143)
  • [25] P. E. Kloeden and J. Lorenz, Stable attracting sets in dynamical systems and in their one-step discretizations, SIAM J. Numer. Anal. 23 (1986), 986-995. MR 859010 (87k:34074)
  • [26] -, A note on multistep methods and attracting sets of dynamical systems, Numer. Math. 56 (1990), 667-673. MR 1031440 (91i:65128)
  • [27] M.-N. Le Roux, Semi-discrétisation en temps pour les équations d'évolution paraboliques lorsque l'opérateur dépend du temps, RAIRO Anal. Numér. 13 (1979), 119-137. MR 533878 (80e:65056)
  • [28] C. Lubich and O. Nevanlinna, On resolvent conditions and stability estimates, BIT 31 (1991), 293-313. MR 1112225 (92h:65145)
  • [29] C. Palencia, Stability of rational multistep approximations of holomorphic semigroups, Math. Comp. 64 (1995), 591-599. MR 1277770 (95g:65079)
  • [30] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sci., vol. 44, Springer, Berlin, 1983. MR 710486 (85g:47061)
  • [31] R. Skeel, Analysis of fixed-stepsize methods, SIAM J. Numer. Anal. 13 (1976), 664-685. MR 0428717 (55:1737)
  • [32] D. Stoffer, General linear methods: connection to one step methods and invariant curves, Numer. Math. 64 (1993), 395-407. MR 1206671 (93m:65098)
  • [33] A. M. Stuart, Nonlinear instability in dissipative finite difference schemes, SIAM Rev. 31 (1989), 191-220. MR 997456 (91b:65110)

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Additional Information

Keywords: Numerical analysis in abstract spaces, approximation of attractors, multistep methods, error bounds and convergence of numerical methods
Article copyright: © Copyright 1995 American Mathematical Society

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