A posteriori error estimates and stopping criteria for space-time domain decomposition for two-phase flow between different rock types
The SMAI Journal of computational mathematics, Volume 5 (2019), pp. 195-227.

We consider two-phase flow in a porous medium composed of two different rock types, so that the capillary pressure field is discontinuous at the interface between the rocks. This is a nonlinear and degenerate parabolic problem with nonlinear and discontinuous transmission conditions on the interface. We first describe a space-time domain decomposition method based on the optimized Schwarz waveform relaxation algorithm (OSWR) with Robin or Ventcell transmission conditions. Complete numerical approximation is achieved by a finite volume scheme in space and the backward Euler scheme in time. We then derive a guaranteed and fully computable a posteriori error estimate that in particular takes into account the domain decomposition error. Precisely, at each iteration of the OSWR algorithm and at each linearization step, the estimate delivers a guaranteed upper bound on the error between the exact and the approximate solution. Furthermore, to make the algorithm efficient, the different error components given by the spatial discretization, the temporal discretization, the linearization, and the domain decomposition are distinguished. These ingredients are then used to design a stopping criterion for the OSWR algorithm as well as for the linearization iterations, which together lead to important computational savings. Numerical experiments illustrate the efficiency of our estimates and the performance of the OSWR algorithm with adaptive stopping criteria on a model problem in three space dimensions. Additionally, the results show how a posteriori error estimates can help determine the free Robin or Ventcell parameters.

Published online:
DOI: 10.5802/smai-jcm.47
Classification: 65M08,  65M15,  65M50,  65M55,  76S05
Keywords: two-phase Darcy flow, discontinuous capillary pressure, finite volume scheme, domain decomposition method, optimized Schwarz waveform relaxation, Robin and Ventcell transmission conditions, linearization, a posteriori error estimate, stopping criteria
Ahmed, Elyes 1; Ali Hassan, Sarah 2; Japhet, Caroline 3; Kern, Michel 4; Vohralík, Martin 4

1 Inria, 2 rue Simone Iff, 75589 Paris, France Current address: Department of Mathematics, University of Bergen, Bergen, Norway
2 Inria, 2 rue Simone Iff, 75589 Paris, France & Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée 2, France
3 Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 93430, Villetaneuse, France
4 Inria, 2 rue Simone Iff, 75589 Paris, France & Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée, France
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     title = {A posteriori error estimates and stopping criteria  for space-time domain decomposition  for two-phase flow between different rock types},
     journal = {The SMAI Journal of computational mathematics},
     pages = {195--227},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Ahmed, Elyes; Ali Hassan, Sarah; Japhet, Caroline; Kern, Michel; Vohralík, Martin. A posteriori error estimates and stopping criteria  for space-time domain decomposition  for two-phase flow between different rock types. The SMAI Journal of computational mathematics, Volume 5 (2019), pp. 195-227. doi : 10.5802/smai-jcm.47. http://archive.numdam.org/articles/10.5802/smai-jcm.47/

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