Doctor of Philosophy (PhD)
In this work we discuss consistency, stability and convergence of rational approximation methods for strongly continuous semigroups on Banach spaces. The Lax-Chernoff theorem shows that in this setting consistency and stability assumptions are necessary to obtain strong uniform convergence of approximation methods. We investigate rational approximation methods for strongly continuous semigroups and their consistency properties, with special emphasis on A-stable methods and Padé-type approximations. In particular, we discuss the stability and convergence properties of these schemes, including the stability of the well-known and widely used Backward-Euler and Crank-Nicolson Schemes. Furthermore, we modify stabilization techniques developed by Hansbo, Larsson, Luskin, Rannacher, Thomée, Wahlbin and others which preserve the order of approximation for non-smooth initial data to be applicable to generators of semigroups which have an H∞-calculus. We introduce time regularization techniques, which - by using a different norm - enable us to obtain optimal convergence rates for time averages for all strongly continuous semigroups not only for smooth but also for non-smooth initial data. Finally, we present new versions of the Trotter-Kato and Lax-Chernoff theorems which yield abstract versions of the Luskin-Rannacher rational stabilization techniques for analytic semigroups in the context of strongly continuous semigroups.
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Flory, Simone, "On the stabilization and regularization of rational approximation schemes for semigroups" (2002). LSU Doctoral Dissertations. 2604.