Multistep approximation of linear sectorial evolution equations

Abstract

This paper concerns the linear multistep approximation of a linear sectorial evolution equation u t = Au on a complex Banach space X. Given a strictly A(a)-stable q-step method of order p whose stability region includes a sectorial region containing the spectrum of the operator A, the corresponding evolution semigroup for the method is C n (hA), n ≥ 0, defined on X q , where C(z) ∈ L(C q ) denotes the one-step map associated with the method. It is shown that for appropriately chosen V, Y: C → C q , based on the principal right and left eigenvectors of C(z), C n (hA) approximates the semigroup V(hA)e nhA Y H (hA) with optimal order p.