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

Summary.

In this paper, we describe a new technique for a posteriori error estimates suitable to parabolic and hyperbolic equations solved by the method of lines. One of our goals is to apply known estimates derived for elliptic problems to evolution equations. We apply the new technique to three distinct problems: a general nonlinear parabolic problem with a strongly monotonic elliptic 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-weighted energy norm. Theory as well as computational results are presented.