Discontinuous and mixed finite elements for two-phase incompressible flow

The simulation of multiphase flow presents several difficulties, including the occurrence of sharp moving fronts when convection is dominating, the need for a good approximation of velocities to calculate the convective terms of the equation, and flow singularities around wells. To handle the first difficulty, the authors propose a Godunov-type higher-order scheme based on a piecewise linear approximation of the saturation associated with a multidimensional slope limiter. With respect to the second, the pressure equation is approximated by means of a mixed-hybrid formulation equivalent to the classic mixed formulation but yielding a positive-definite linear system. To solve the third difficulty, the authors introduce macroelements around wells. Numerical experiments illustrate the capabilities of the method.