Multistep approximation of linear sectorial evolution equations
Adrian T. Hill · IMA Journal of Numerical Analysis · 2004
This paper concerns the linear multistep approximation of a linear sectorial evolution equation ut = Au on a complex Banach space X. Given a strictly A(α)-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 Cn(hA), n ≥0, defined on Xq, where C(z) ∈ L (Cq) denotes the one-step map associated with the method. It is shown that for appropriately chosen V, Y: C → Cq, based on the principal right and left eigenvectors of C(z), Cn(hA) approximates the semigroup V(hA)enhAYH(hA) with optimal order p.