Section: New Results

A posteriori error estimates

Participant : Martin Vohralík.

In [2] , we have been able to derive an a posteriori error estimate for the numerical approximation of the two-phase flow problem. This is a cornerstone model problem for porous media, describing the flow of two immiscible and incompressible fluids. We take into account the capillary pressure, whence the model features such difficulties as coupling of partial differential equations with algebraic constraints, strong nonlinearities, degeneracy (disappearance of the diffusion term), advection dominance and consequent forming of sharp evolving fronts, or highly nonlinear and very badly conditioned systems of algebraic equations. Our analysis covers a large class of spatial discretizations in a unified setting, with fully implicit time stepping. We also show how the different error components, namely the spatial discretization error, the temporal discretization error, the linearization error, and the algebraic solver error can be distinguished and estimated separately. This gives rise to efficient adaptive stopping criteria, enabling to spare many useless iterations. The practical impact of our results is that even for this complicated model problem, the overall error committed in a numerical approximation can be fully controlled and, moreover, the simulation time can be reduced by factors typically of an order of magnitude. This result has then been extended in [4] to the compositional model of multiphase Darcy flow, where an arbitrary number of phases can be present, and where each phase can be composed of several components. Later, in [12] , still a possible dependence on the temperature has been added. The last two references also contain convincing numerical illustrations on real-life reservoir engineering examples.