Fully Discrete Approximation Methods for Parabolic Poblems with Nonsmooth Initial Data

Fully discrete approximation methods are studied for parabolic boundary value problems evolving from nonsmooth (and noncompatible) initial data. Galerkin methods are used for the spatial discretization, and single step or backwards differentiation multistep methods are used in time. The parabolic equation is assumed to be linear, but the coefficients are allowed to depend on space and time. An error estimate is given for the combined space-time discretization scheme which contains the expected optional order space and time error contributions. However, another term appears which limits the accuracy of the method to second order (or fractionally better) in time if the time stepping method is of third or higher order. This latter term arises from the time dependence of the equation’s highest order coefficients, and hence does not appear in analyses of problems with only spatially dependent coefficients for which optimal order estimates are known to hold. It is shown through more detailed analysis of a specific problem, and the formulation of a lower error bound, that the error estimate is indeed sharp. Thus, it follows that standard higher order time stepping schemes cannot be expected to perform well for parabolic problems with time dependent coefficients in situations when nonsmooth initial data are encountered.