An A Posteriori Error Estimator for a Finite Volume Discretization of the Two-phase Flow

We derive a posteriori error estimates for a multi-point finite volume discretization of the two-phase Darcy problem. The proposed estimators yield a fully computable upper bound for the selected error measure. The estimate also allows to distinguish, estimate separately, and compare the linearization and algebraic errors and the time and space discretization errors. This enables, in particular, to design a discretization algorithm so that all the sources of error are properly balanced. Namely, the linear and nonlinear solvers can be stopped as soon as the algebraic and linearization errors drop to the level at which they do not affect to the overall error. This can lead to significant computational savings, since performing an excessive number of unnecessary iterations can be avoided. Similarly, the errors in space and in time can be equilibrated by time step and local mesh adaptivity.