A convergent mass conservative numerical scheme based on mixed finite elements for two-phase flow in porous media
This work provides a theoretically grounded numerical method for simulating two-phase flow in porous media, which is important for applications like groundwater and oil reservoir modeling.
The paper presents a mass conservative numerical scheme for two-phase flow in porous media using mixed finite elements, with rigorous convergence proofs and error estimates. Numerical results confirm the theoretical findings.
In this work we present a mass conservative numerical scheme for two-phase flow in porous media. The model for flow consists on two fully coupled, non-linear equations: a degenerate parabolic equation and an elliptic equation. The proposed numerical scheme is based on backward Euler for the temporal discretization and mixed finite element method (MFEM) for the discretization in space. Continuous, semi-discrete (continuous in space) and fully discrete variational formulations are set up and the existence and uniqueness of solutions is discussed. Error estimates are presented to prove the convergence of the scheme. The non-linear systems within each time step are solved by a robust linearization method. This iterative method does not involve any regularization step. The convergence of the linearization scheme is rigorously proved under the assumption of a Lipschitz continuous saturation. The case of a Hölder continuous saturation is also discussed, a rigorous convergence proof being given for Richards' equation. Numerical results are presented to sustain the theoretical findings.