Fully reliable error control for first-order evolutionary problems

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 solution approximation 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 minimisation of the majorant on each of discretisation time-cylinders (time-slabs), the effectiveness of this approach to error estimation 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 demonstrates the efficiency of proposed error control method. The numerical results obtained in this paper rely on the implementation carried out using open source software, which allows formulating the problem in a weak setting. To work in a variational framework appears to be very natural for functional error estimates due to their derivation method. The search for the optimal parameters for the majorant is done by a global functional minimisation, which to the authors’ knowledge is the first work using this technique in an evolutionary framework.