A posteriori error estimates based on multilevel decompositions with an iterative solver on the coarsest level

Multilevel methods represent a powerful approach to the numerical solution of partial differential equations. The multilevel structure can also be used to construct estimates for the total and algebraic errors of the computed approximations. This paper deals with residual-based error estimates that rely 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 crucial. It can significantly affect both the efficiency (accuracy) of the overall error estimates and their robustness with respect to the size of the coarsest-level problem. We propose a new approximation of the coarsest-level term 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.