International audienceIn this paper we derive a posteriori error estimates for the compositional model of multiphase Darcy flow in porous media, consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations. We show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators. We then decompose the estimators into the space, time, linearization, and algebraic error components. This allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error. Moreover, the spatial and temporal error components can be balanced by time step and space mesh adaptation. Our analysis applies to a broad class of standard numerical methods, and is independent of the linearization and of the iterative algebraic solvers employed. We exemplify it for the two-point finite volume method with fully implicit Euler time stepping, the Newton linearization, and the GMRes algebraic solver. Numerical results on two real-life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria, already on fixed meshes, without any noticeable loss of precision

We consider in this paper the time-dependent two-phase Stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward Euler temporal discretizations. Regularization of the enthalpy-temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered. Our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the L 2 (L 2 ) error of the temperature and the L 2 (H −1 ) error of the enthalpy. Moreover, they allow to distinguish the space, time, regularization, and linearization error components. An adaptive algorithm is proposed, which ensures computational savings through the online choice of a sufficient regularization parameter, a stopping criterion for the linearization iterations, local space mesh refinement, time step adjustment, and equilibration of the spatial and temporal errors. We also prove the efficiency of our estimate. Our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization. As an example, we apply it to the vertex-centered finite volume (finite element with mass lumping and quadrature) and Newton methods. Numerical results illustrate the effectiveness of our estimates and the performance of the adaptive algorithm.MSC: 65N08, 65N15, 65N50, 80A22 IntroductionThe two-phase Stefan problem models a phase change process which is governed by the Fourier law, c.f. Friedman [22]. The two phases, typically solid and liquid, are separated by a moving interface, whose motion is governed by the so-called Stefan condition., be an open bounded polygonal or polyhedral domain, not necessarily convex, and let T > 0. The mathematical statement of the problem is as follows: given an initial enthalpy u 0 and a source function f , find the enthalpy u such that(1.1c) * This work was supported by the ERT project "Enhanced oil recovery and geological sequestration of CO 2 : mesh adaptivity, a posteriori error control, and other advanced techniques" (LJLL/IFPEN) † daniele.di-pietro@univ-montp2.fr ‡