Approximation of the Semigroup of a Linear Age-dependent Population Model with Spatial Diiusion Running Head: Age-dependent Population Diiusion Mail Proofs To

In this paper a variational version of the Trotter-Kato theorem is applied to an innnitesimal generator A of a C 0-semigroup in a Hilbert space that is assumed to be the sum of two linear operators A and B deened on the intersection of their domains, where A ? !I is m-dissipative for some ! > 0 and B is the generator of an analytic semigroup associated with a bounded V-coercive sesquilinear form. An approximation of the semigroup generated by A is obtained from given approximations of the sesquilinear forms appertaining to A and B. This result is applied to the semigroup generated by an age-dependent population model with spatial diiusion, and numerical examples are presented to demonstrate feasibility of the scheme. 2 1. Introduction Approximation schemes for the semigroups of age-, size-and space-structured population models have been considered in several papers, where the proof of convergence uses one or the other version of the Trotter-Kato theorem in semigroup theory (see, for instance, 1], 2], 7], 10] and 11]). In 9] and 10] variational formulations of the Trotter-Kato theorem were derived. The characteristic feature of this approach is that instead of approximating the generator one rather approximates the associated sesquilinear form rst and then gets the approximating generators from the approximating sesquilinear forms. In many cases the proof of consistency for a variational Trotter-Kato scheme is considerably simpler compared to the one for an ordinary Trotter-Kato scheme. In 9] a special type of variational Trotter-Kato theorem for parabolic semigroups in a Hilbert space whose innnitesimal generator is associated with a V-coercive sesquilinear form was developed.