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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
DOI: https://doi.org/10.1090/S0025-5718-1995-1297470-2
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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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1995-1297470-2
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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