The accuracy estimates of the Cayley transform method for the abstract Cauchy problem

Abstract

We obtain the accuracy estimates of the Cayley transform method for solving the initial value problem for a homogeneous first-order differential equation with an unbounded operator coefficient in a Hilbert space. In the case of finite (in some sense) smoothness of the initial vector, our method has a power-law rate of convergence and, moreover, the rate automatically depends on this regularity (i.e. the Cayley transform method is a method without saturation of accuracy). If the initial vector is infinitely smooth, then our method is exponentially convergent. In addition, we substantiate that the estimates are unimprovable in the order of N (the discretization parameter N characterizes the number of summands in the partial sum of the approximate solution).