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

Hill, Adrian T.; Süli, Endre

doi:10.1090/S0025-5718-1995-1297470-2

Upper semicontinuity of attractors for linear multistep methods approximating sectorial evolution equations
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by Adrian T. Hill and Endre Süli PDF

Math. Comp. 64 (1995), 1097-1122 Request permission

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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