A posteriori error estimates for a compositional two-phase flow with nonlinear complementarity constraints

Abstract

In this work, we develop an a-posteriori-steered algorithm for a compositional two-phase flow with exchange of components between the phases in porous media. As a model problem, we choose the two-phase liquid-gas flow with appearance and disappearance of the gas phase formulated as a system of nonlinear evolutive partial differential equations with nonlinear complementarity constraints. The discretization of our model is based on the backward Euler scheme in time and the finite volume scheme in space. The resulting nonlinear system is solved via an inexact semismooth Newton method. The key ingredient for the a posteriori analysis are the discretization, linearization, and algebraic flux reconstructions allowing to devise estimators for each error component. These enable to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver whenever the corresponding error components do not affect significantly the overall error. Numerical experiments are performed using the Newton-min algorithm as well as the Newton-Fischer-Burmeister algorithm in combination with the GMRES iterative linear solver to show the efficiency of the proposed adaptive method.

Semismooth and smoothing Newton methods for nonlinear systems with complementarity constraints: Adaptivity and inexact resolution

“…The formulation (7.1) allows to model the transition from a single-phase flow to a two-phase flow during the appearance and disappearance of the gas phase and vice versa. As an example, a detailed finite volume discretization can be found in [7,Section 3.2]. The first 2N lines of system (7.1) can be written globally as…”

A posteriori estimates distinguishing the error components and adaptive stopping criteria for numerical approximations of parabolic variational inequalities

“…Problem (1.2) belongs to the wide class of parabolic variational inequalities of the first kind, see Glowinski [1] and Lions [2] for a general introduction. These have attracted a recent interest in a wide variety of applications; we mention the obstacle problems in mechanics [2,3,4], the problems in modeling pricing of American options [5,6], the Stefan problem [7], the CO 2 sequestration process [8], the mould filling [9], and the underground storage of radioactive waste [10]. Though the existence and uniqueness of a weak solution u ∈ K t g for (1.2) is classical, see [2] and the references therein, the numerical analysis of parabolic variational inequalities is very challenging.…”