2013/06/12 by Bilal Saad, Mazen Saad, Saad, Bilal +1
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1306.2867
openalex publication_date 2013/06/12 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We propose and analyze a combined finite volume--nonconforming finite element\nscheme on general meshes to simulate the two compressible phase flow in porous\nmedia. The diffusion term, which can be anisotropic and heterogeneous, is\ndiscretized by piecewise linear nonconforming triangular finite elements. The\nother terms are discretized by means of a cell-centered finite volume scheme on\na dual mesh, where the dual volumes are constructed around the sides of the\noriginal mesh. The relative permeability of each phase is decentred according\nthe sign of the velocity at the dual interface. This technique also ensures the\nvalidity of the discrete maximum principle for the saturation under a non\nrestrictive shape regularity of the space mesh and the positiveness of all\ntransmissibilities. Next, a priori estimates on the pressures and a function of\nthe saturation that denote capillary terms are established. These stabilities\nresults lead to some compactness arguments based on the use of the Kolmogorov\ncompactness theorem, and allow us to derive the convergence of a subsequence of\nthe sequence of approximate solutions to a weak solution of the continuous\nequations, provided the mesh size tends to zero. The proof is given for the\ncomplete system when the density of the each phase depends on the own pressure.\n