A posteriori error estimates including algebraic error: computable upper bounds and stopping criteria for iterative solvers

For the finite volume discretization of a second-order elliptic model problem, we derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system. We show that the algebraic error can be bounded by constructing an equilibrated Raviart--Thomas--Nédélec discrete vector field whose divergence is given by a proper weighting of the residual vector. Next, claiming that the discretization error and the algebraic one should be in balance, we construct stopping criteria for iterative algebraic solvers. An attention is paid, in particular, to the conjugate gradient method which minimizes the energy norm of the algebraic error. Using this convenient balance, we also prove the efficiency of our a posteriori estimates, i.e., we show that they also represent a lower bound, up to a generic constant, for the overall energy error. A local version of this result is also stated. This makes our approach suitable for adaptive mesh refinement which also takes into account the algebraic error. Numerical experiments illustrate the proposed estimates and construction of efficient stopping criteria for algebraic iterative solvers.