On one approach to a posteriori error estimates for evolution problems solved by the method of lines

In this paper, we describe a new technique for a posteriori error estimates suitable to parabolic and hyperbolic equations solved by the method of lin es. One of our goals is to apply known estimates derived for elliptic probl ems to evolution equations. We apply the new technique to three distinct pr oblems: a general nonlinear parabolic problem with a strongly monotonic ell iptic operator, a linear nonstationary convection-diffusion problem, and a linear second order hyperbolic problem. The error is measured with the aid of the L-2-norm in the space-time cylinder combined with a special time-wei ghted energy norm. Theory as well as computational results are presented.