Space-time domain decomposition for two-phase flow between different rock types

This paper is dedicated to Professor Mary F. Wheeler on the occasion of her 80thbirthday.The authors warmly thank Cl ément Cancès and Robert Eymard for enlightening discussions that helped usto carry out the proof of Theorem 4.1.International audienceIn previous work (E. Ahmed, S. Ali Hassan, C. Japhet, M. Kern, and M. Vohralík, "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, 5 (2019), pp. 195{227, https://doi.org/10.5802/smai-jcm.47) a space-time domain decomposition method was proposed for two-phase ow in a porous medium composed of two different rock types, so that the capillary pressure field is discontinuous at the interface between the rocks. For this nonlinear and degenerate parabolic problem, with nonlinear and discontinuous transmission conditions on the interface, the Optimized Schwarz waveform relaxation method (OSWR) with Robin or Ventcell transmission conditions was considered. A guaranteed and fully computable a posteriori error estimate was derived, which in particular took into account the domain decomposition error. In this paper we provide a mathematical and numerical analysis of this space{time domain decomposition method in the Robin case. Complete numerical approximation is achieved by a finite volume scheme in space and the lowest order discontinuous Galerkin method in time. We prove the existence of a weak solution of the two-phase ow subdomain problem with Robin boundary conditions by analyzing the convergence of the finite volume scheme. The domain decomposition algorithm is based on thesolution of space-time nonlinear subdomain problems over the whole time interval, allowing for different time steps in dfferent parts of the domain, adapted to the physical properties of each subdomain, and we show that such an algorithm is well-defined. Numerical experiments on three-dimensional problems with different rock types illustrate the performance of the domain decomposition method