Stability of W-methods with applications to operator splitting and to geometric theory

Abstract

We analyze the stability properties of W-methods applied to the parabolic initial value problem u′+ Au= Bu. We work in an abstract Banach space setting, assuming that A is the generator of an analytic semigroup and that B is relatively bounded with respect to A. Since W-methods treat the term with A implicitly, whereas the term involving B is discretized in an explicit way, they can be regarded as splitting methods. As an application of our stability results, convergence for nonsmooth initial data is shown. Moreover, the layout of a geometric theory for discretizations of semilinear parabolic problems u′+ Au= f( u) is presented.