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Mathematics of Computation

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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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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