Approximation of Semigroups and Related Operator Functions by Resolvent Series

Abstract. We consider the approximation of semigroups e and of the functions φj(τA) that appear in exponential integrators by resolvent series. The interesting fact is that the resolvent series expresses the operator functions e and φj(τA), respectively, in efficiently computable terms. This is important for semigroups, where the new approximation is different from well-known approximations by rational functions, as well as for the application of exponential integrators, which are currently of high interest and which are usually studied in a semigroup setting on Banach spaces. The approximation of the operator functions φj(τA) in a general strongly continuous semigroup setting has not been discussed in the literature so far, while this is crucial for an application of these integrators with unbounded operators or bounded operators (like discretization matrices) with large norm and eigenvalues somewhere in the left half plane.