A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows

Martin Vohralík (1) , Mary Wheeler (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 ICES - Institute for Computational Engineering and Sciences [Austin]

This paper develops a general abstract framework for a posteriori estimates of the dual norm of the residual for immiscible incompressible two-phase flows in porous media. Our estimators allow to estimate separately the different error components, namely the spatial discretization error, the temporal discretization error, the linearization error, the iterative coupling error, and the algebraic solver error. We propose an adaptive algorithm wherein the different iterative procedures (iterative linearization, iterative coupling, iterative solution of linear systems) are stopped when the corresponding errors do not affect significantly the overall error, and wherein the spatial and temporal errors are equilibrated. Consequently, important computational savings may be achieved while guaranteeing a user-given precision. The developed framework covers fully implicit, implicit pressure-explicit saturation, or iterative coupling formulations; conforming spatial discretization schemes such as the vertex-centered finite volume method or the finite element method and nonconforming spatial discretization schemes such as the cell-centered finite volume method, the mixed finite element method, or the discontinuous Galerkin method; linearizations such as the Newton or the fixed-point one; and general linear solvers.

Mots clés: two-phase flow; a posteriori error estimate; general framework; discretization error; linearization error; iterative coupling error; algebraic solver error

Domaines: Analyse numérique [math.NA]

Martin Vohralík, Mary Wheeler. A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows. 2011. ⟨hal-00633594v1⟩