Fabricio S. Sousa

High-order ALE schemes for incompressible capillary flows

The spatial discretization of problems with moving boundaries is considered, incorporating the temporal evolution of not just the mechanical variables, but also of the nodal positions of the moving mesh. The outcome is a system of Differential-Algebraic Equations (DAE) of index 2 or, in the case of inertialess flow, just 1. From the DAE formulation it its possible to define strategies to build schemes of arbitrary accuracy. We introduce here several schemes of order 2 and 3 that avoid the solution of a nonlinear system involving simultaneously the mechanical variables and the geometrical ones. This class of schemes has been the one adopted by the majority of practitioners of Computational Fluid Dynamics up to now. The proposed schemes indeed achieve the design accuracy, and further show stability restrictions that are not significantly more severe than those of popular first order schemes. The numerical experimentation is performed on capillary problems, discretized by both div-stable ( P 2 / P 1 , P 1 + / P 1 ) and equal-order ( P 1 / P 1 , stabilized) finite elements, and incorporating surface tension and triple (contact) line effects.

The Multiscale Robin Coupled Method for flows in porous media

Abstract A multiscale mixed method aiming at the accurate approximation of velocity and pressure fields in heterogeneous porous media is proposed. The procedure is based on a new domain decomposition method in which the local problems are subject to Robin boundary conditions. The domain decomposition procedure is defined in terms of two independent spaces on the skeleton of the decomposition, corresponding to interface pressures and fluxes, that can be chosen with great flexibility to accommodate local features of the underlying permeability fields. The well-posedness of the new domain decomposition procedure is established and its connection with the method of Douglas et al. (1993) [12] , is identified, also allowing us to reinterpret the known procedure as an optimized Schwarz (or Two-Lagrange-Multiplier) method. The multiscale property of the new domain decomposition method is indicated, and its relation with the Multiscale Mortar Mixed Finite Element Method (MMMFEM) and the Multiscale Hybrid-Mixed (MHM) Finite Element Method is discussed. Numerical simulations are presented aiming at illustrating several features of the new method. Initially we illustrate the possibility of switching from MMMFEM to MHM by suitably varying the Robin condition parameter in the new multiscale method. Then we turn our attention to realistic flows in high-contrast, channelized porous formations. We show that for a range of values of the Robin condition parameter our method provides better approximations for pressure and velocity than those computed with either the MMMFEM and the MHM. This is an indication that our method has the potential to produce more accurate velocity fields in the presence of rough, realistic permeability fields of petroleum reservoirs.