A combined finite volume--nonconforming finite element scheme for compressible two phase flow in porous media

TL;DR

This paper introduces and analyzes a combined finite volume--nonconforming finite element scheme for simulating two-phase compressible flow in porous media, ensuring stability, convergence, and physical constraints on general meshes.

Contribution

It develops a novel combined discretization method that handles anisotropic diffusion and guarantees maximum principles and convergence for two-phase flow models.

Findings

Proves stability and maximum principle preservation.

Establishes convergence of the scheme to weak solutions.

Handles anisotropic, heterogeneous diffusion in porous media.

Abstract

We propose and analyze a combined finite volume--nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. The relative permeability of each phase is decentred according the sign of the velocity at the dual interface. This technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities. Next, a priori estimates on the pressures and a function of the saturation that denote capillary terms are established. These stabilities results lead to some compactness arguments based on the use of the Kolmogorov compactness theorem, and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations, provided the mesh size tends to zero. The proof is given for the complete system when the density of the each phase depends on the own pressure.