Simplified Weak Galerkin Discretization for Quasi-Incompressible Cahn-Hilliard-Darcy Equations: Algorithm and error analysis

This work extends the Simplified Weak Galerkin (SWG) finite element framework to the quasi-incompressible Cahn-Hilliard-Darcy (qCHD) system, which models two-phase flows in porous media. The proposed formulation handles the coupled nonlinearities through a stabilizer and numerical flux design that maintains accuracy while ensuring computational efficiency. For temporal discretization, we construct a linearly implicit scheme that treats nonlinear couplings explicitly while maintaining stability through an implicit treatment of diffusive terms. The spatial discretization captures the intricate interface dynamics and pressure-velocity constraints inherent in two-phase porous media flows. The method demonstrates robustness across various flow regimes while maintaining the structure-preserving properties of the SWG framework. Mathematics Subject Classification: 65M60, 76M10, 76S05, 35Q35.