  This work is focused on the application of functional-type a posteriori error
estimates and corresponding indicators to a class of time-dependent problems.
We consider the algorithmic part of their derivation and implementation and
also discuss the numerical properties of these bounds that comply with obtained
numerical results. This paper examines two different methods of approximate
solution reconstruction for evolutionary models, i.e., a time-marching
technique and a space-time approach. The first part of the study presents an
algorithm for global minimization of the majorant on each of discretization
time-cylinders (time-slabs), the effectiveness of this algorithm is confirmed
by extensive numerical tests. In the second part of the publication, the
application of functional error estimates is discussed with respect to a
space-time approach. It is followed by a set of extensive numerical tests that
demonstrate the efficiency of proposed error control method.
