In this dissertation, we formulate a pressure-correction projection algorithm, in conjunction with the interior penalty discontinuous Galerkin scheme for time and space discretization to build a single-phase incompressible Navier–Stokes simulator and a two-phase Cahn–Hilliard–Navier–Stokes simulator. The method is a decoupled algorithm, which is especially convenient for large-scale 3D numerical simulations in complex geometry, such as in porous structures obtained from microtomography scanning. The simulators we implemented are robust. The numerical experiment results have been validated on a series of realistic physical problems and exhibit the potential for computing effective properties of single/two-phase flow such as permeability and saturation.

Theoretical analysis of the numerical methods for solving multiphase flow model will also contribute to the understanding of the complex multi-scale fluid system from the mathematical point of view. In this dissertation, we also analyze a non-symmetric interior penalty discontinuous Galerkin scheme for solving the mixed form of the Cahn–Hilliard equation and a symmetric interior penalty discontinuous Galerkin scheme for solving the Cahn–Hilliard–Navier–Stokes equations. We prove several numerical properties for these numerical schemes, including unique solvability, stability analysis, and error analysis.