  We propose a method to integrate dissipative PDEs rigorously forward in time
with the use of Finite Element Method (FEM). The technique is based on the
Galerkin projection on the FEM space and estimates on the residual terms. The
proposed approach is illustrated on a periodically forced one-dimensional
Burgers equation with Dirichlet conditions. For two particular choices of the
forcing we prove the existence of the periodic globally attracting trajectory
and give precise bounds on its shape.
