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An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow

Authors: Clément Cancès, Iuliu Sorin Pop and Martin Vohralík
Journal: Math. Comp. 83 (2014), 153-188
MSC (2010): Primary 65M15, 76S05, 76T99, 65M08
Published electronically: June 28, 2013
MathSciNet review: 3120585
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Abstract: In this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium. We take into account the capillary pressure, which leads to a coupled system of two equations: parabolic and elliptic. The parabolic equation may become degenerate, i.e., the nonlinear diffusion coefficient may vanish over regions that are not known a priori. We first show that, under appropriate assumptions, the energy-type norm differences between the exact and the approximate nonwetting phase saturations, the global pressures, and the Kirchhoff transforms of the nonwetting phase saturations can be bounded by the dual norm of the residuals. We then bound the dual norm of the residuals by fully computable a posteriori estimators. Our analysis covers a large class of conforming, vertex-centered finite volume-type discretizations with fully implicit time stepping. As an example, we focus here on two approaches: a ``mathematical'' scheme derived from the weak formulation, and a phase-by-phase upstream weighting ``engineering'' scheme. Finally, we show how the different error components, namely the space discretization error, the time discretization error, the linearization error, the algebraic solver error, and the quadrature error can be distinguished and used for making the calculations efficient.

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Additional Information

Clément Cancès
Affiliation: LJLL – UPMC Paris 06, Boite Courrier 187, 4 place Jussieu, 75005 Paris, France

Iuliu Sorin Pop
Affiliation: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB, Eindhoven, the Netherlands

Martin Vohralík
Affiliation: LJLL – UPMC Paris 06, Boite Courrier 187, 4 place Jussieu, 75005 Paris, France
Address at time of publication: INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France

Received by editor(s): September 13, 2011
Received by editor(s) in revised form: April 25, 2012
Published electronically: June 28, 2013
Additional Notes: This work was partly supported by the Groupement MoMaS (PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN) and by the ERT project “Enhanced oil recovery and geological sequestration of $\mathrm{CO}_{2}$: mesh adaptivity, a posteriori error control, and other advanced techniques” (LJLL/IFPEN)
Article copyright: © Copyright 2013 American Mathematical Society

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