Analysis of the Properties of Mixed Method Discrete Analogues for Solving the Darcy Problem

This paper is devoted to mathematical modeling of the fluid filtration process in a porous geological medium under pressure. Two variants of a mixed discontinuous Galerkin formulation with different stabilization methods for solving the Darcy problem with a tensor permeability coefficient of the medium are considered. The construction of discrete analogues of these formulations is carried out using a hierarchical basis system for velocity and a discontinuous basis for pressure in specialized functional spaces. The analysis of the properties of matrices of systems of linear algebraic equations is carried out on the framework of a singular value decomposition of these matrices. The authors have developed and verified a software package in the C++ programming language that accepts as input a grid partition of the computational domain in the ".msh" format, obtained using a finite element mesh generator Gmsh. The implemented software package generates the matrices of the system of linear algebraic equations using the constructed basis systems and then performs a singular value decomposition, which results in an ordered vector of singular numbers. This vector is displayed as a result for further analysis.