Smoothing properties and approximation of time derivatives for parabolic equations: constant time steps

Journal article, 2003

We study smoothing properties and approximation of time derivatives for time discretization schemes with constant time steps for a homogeneous parabolic problem formulated as an abstract initial-value problem in a Banach space. The time stepping schemes are based on using rational functions r(z) e–z which are A()-stable for suitable [0, /2] and satisfy |r()| < 1, and the approximations of time derivatives are based on using difference quotients in time. Both smooth and non-smooth data error estimates of optimal order for the approximation of time derivatives are proved. Further, we apply the results to obtain error estimates of time derivatives in the supremum norm for fully discrete methods based on discretizing the spatial variable by a finite-element method.