  Multilevel methods represent a powerful approach in numerical solution of
partial differential equations. The multilevel structure can also be used to
construct estimates for total and algebraic errors of computed approximations.
This paper deals with residual-based error estimates that are based on
properties of quasi-interpolation operators, stable-splittings, or frames. We
focus on the settings where the system matrix on the coarsest level is still
large and the associated terms in the estimates can only be approximated. We
show that the way in which the error term associated with the coarsest level is
approximated is substantial. It can significantly affect both the efficiency
(accuracy) of the overall error estimates and their robustness with respect to
the size of the coarsest problem. The newly proposed approximation of the
coarsest-level term is based on using the conjugate gradient method with an
appropriate stopping criterion. We prove that the resulting estimates are
efficient and robust with respect to the size of the coarsest-level problem.
Numerical experiments illustrate the theoretical findings.
