FINITE ELEMENT CENTER PREPRINT 2002–02 Approximation of time derivatives for parabolic equations in Banach space: variable time steps

We study smoothing properties and approximation of time derivatives for time discretization schemes with variable time steps for a homogeneous parabolic problem formulated as an abstract initial value problem in a Banach space. The time stepping methods are based on using rational functions r(z) e ?z which are A()-stable for suitable 2 (0; =2] and satisfy jr(1)j < 1. First and second order approximations of time derivatives based on using diierence quotients are considered. Smoothing properties are derived and error estimates are established under the so called increasing quasi-quasiuniform assumption on the time steps.