Analysis of Finite Element Discretisation Schemes for Multi-Phase Darcy Flow
We demonstrate that care must be taken in a finite-element discretisation of multi-phase compressible Darcy flow, otherwise constraints of non-negativity of fluid mass can become violated. Generalising a technique pioneered by Dalen, a lumped, finite-difference-inspired approximation with upstream weighting is described. This is numerically cheap, physically elegant, and the fluid mass at all nodes remains non-negative.
D. P. Adhikary and A. H. Wilkins, "Analysis of Finite Element Discretisation Schemes for Multi-Phase Darcy Flow", Defect and Diffusion Forum, Vol. 318, pp. 33-40, 2011