Title page for ETD etd-07142005-114344


Type of Document Dissertation
Author McAllister, Sarah Campbell
Author's Email Address smcall1@lsu.edu
URN etd-07142005-114344
Title Error Estimates for Stabilized Approximation Methods for Semigroups
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Frank Neubrander Committee Chair
Peter Wolenski Committee Member
R.A. Litherland Committee Member
Robert Perlis Committee Member
Stephen Shipman Committee Member
William Metcalf Dean's Representative
Keywords
  • stabilization
  • error estimates
  • time discretization
  • evolution equations
  • fractional powers
Date of Defense 2005-07-08
Availability unrestricted
Abstract
In this work we analyze error estimates for rational approximation methods, and their stabilizations, for strongly continuous semigroups. Chapter 1 consists of a brief survey of time discretization methods for semigroups. In Chapter 2, we demonstrate a new method for obtaining convergent approximations in the absence of stability for strongly continuous semigroups with arbitrary initial data. In Section 2.2, we state the stabilization result in more general form and show that this method can be used to improve known error estimates by a magnitude of up to one half for smooth initial data. In Section 2.3, we give concrete examples of some of these stabilizers. Section 2.4 concerns abstract stabilization results, including stabilized Trotter-Kato and Lax-Chernoff theorems. In Chapter 3, we use numerical quadrature formulas for Banach space valued functions in order to approximate semigroups that can be represented via

the Hille-Phillips functional calculus. In particular, we find error estimates for our

approximation method for the semigroup generated by the square root of a semigroup

generator.

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