On Approximation by Discrete Semigroups

Abstract

The present paper deals with the problem of approximation of a continuous parameter semigroup T( t), t > 0 on a Banach space X by means of a sequence of discrete parameter semigroups ( F k n ), where F n is a bounded operator on a Banach space X n , n ∈ N, and where ( X n ) and X are related in some appropriate sense. This problem arises, e.g., when numerical methods are used to approximate solutions of initial boundary value problems in PDEs. The results obtained here present a new set of tests for convergence of discrete semigroups, which are different from those in (E. Görlich and D. Pontzen, Tôhuku Math. J. (2) 34, No.4 (1982), 539-552). Theorem 2 and its corollaries extend the earlier results on this point.