This work is devoted to fully reliable a posteriori error analysis for a class of evolutionary problems and some questions emerging in relation to it. The first articles in this collection are concerned with theoretical and numerical analysis, efficient and robust implementation of the functional type a posteriori error estimates and indicators for the nonlinear Cauchy problem, and time-dependent reaction-diffusion initial-boundary value problems of parabolic type. The last part of the study is dedicated to computable and sharp upper bounds of constants in Poincaré-type inequalities for functions with zero mean on the boundary (or a measurable part of it) on non-degenerate triangles and tetrahedrons. These sharp upper bounds are crucial for quantitative analysis of problems generated by differential equations, where numerical approximations are typically constructed with the help of simplicial meshes and become particularly useful in the implementation of the functional error majorants applied for the problems with a decomposed domain. The error estimates presented in this thesis are explicitly computable and guaranteed. The two-sided functional type error bounds hold for all conforming approximations, do not depend on any mesh discretization parameters, and only contain global and local constants in Poincaré inequalities. Extensive numerical experiments, performed alongside with theoretical findings, provide results, which confirm the efficiency and reliability of the error estimates and robustness of the indicators they comprise. For numerical implementation we use MATLAB and The FEniCS Project (with Python).