Nonsmooth Data Error Estimates with Applications to the Study of the Long-time Behavior of Finite Element Solutions of Semilinear Parabolic Problems

A rather general semilinear parabolic problem is studied together with its spatially semidiscrete nite element approximation. Both problems are formulated within the framework of nonlinear semigroups in the Sobolev space H 1 ((). The main result is an error estimate for solutions with initial data in H 1 ((), valid during an arbitrary nite time interval. The proof is based on the semigroup formulation and is rather elementary. A completely discrete scheme based on the backward Euler method for time discretization is analyzed in an analogous way. This type of error estimate is a key ingredient in recent work by several authors concerning the long-time behavior of nite element solutions. Two examples of this kind are presented: the upper semicontinuity of attractors and a long-time error bound for exponentially stable solutions.