Stabilized approximations of strongly continuous semigroups

Abstract

This paper introduces stabilization techniques for intrinsically unstable, high accuracy rational approximation methods for strongly continuous semigroup. The methods not only stabilize the approximations, but improve their speed of convergence by a magnitude of up to 1/2. PreliminariesThe Lax-Richtmyer Equivalence Theorem and the Chernoff Product Formula play an important role in the analysis of approximation methods for semigroups. One of the main ingredients in both results is the stability of the approximation method under consideration. This paper deals with variants of these theorems that cover stabilization techniques for intrinsically unstable approximation methods. To put the results into perspective, recall the statement of the LaxRichtmyer Equivalence Theorem. Suppose that A generates a strongly continuous semigroup T (·) on a Banach space X and let {V (t); t ∈ [0, τ ]} be an approximation scheme of bounded linear operators with V (0) = I that satisfies the consistency conditionfor all x in a set D ⊂ D(A) (the domain of A) that is dense in X. Then it was shown by Lax and Richtmyer in 1956 [19] (with a stronger consistency condition) and in final form by Chernoff in 1974 [6] that the following statements are equivalent (see also [9]):

Rational inversion of the Laplace transform

“…Rannacher [31], and in the final form Hansbo [15] proposed a stabilization technique for unstable rational approximations by first applying a stable lower order approximation with r(∞) = 0 and then combining the smoothing property of the lower order scheme with the improved accuracy of the higher order approximation scheme. The stabilization of rational approximation schemes for non-analytic strongly continuous semigroups was investigated by McAllister and Neubrander in [28]. For applications of these stabilization techniques to Laplace transform inversions, see [29].…”

“…For u ∈ C ub (Σ θ , X) ∩ H(Σ θ , X) the results can be improved by using Hansbo's stabilization methods [15]; for u, u (1) ∈ C b (R + , X) the results can be improved by stabilizing the Crank-Nicolson scheme using the methods in [28] (see [29]). …”

An adjoint control method for initial condition identification of the Abstract Cauchy problem

“…using the relation (8) and equation (14). Further analysis of the method is detailed below, including the error analysis in Theorem 2.1.…”

“…where the degree of P, Q is not more than m, n respsectively. Higher order Padé approximations of semigroups are discussed in detail in the paper [14].…”