The expanded upwind-mixed multi-step method for the miscible displacement problem in three dimensions

The expanded upwind-mixed multi-step method is presented for the miscible displacement in three dimensions. The pressure is approximated by a mixed finite element method and the concentration by a method which upwinds the convection and incorporates diffusion using an expanded mixed finite element method. The method improves upon previous methods as follows: first, it can keep conservation of mass; second, the multi-step method has the higher precision and reduces the work involved in the actual computation; third, the method does not need the Ω-periodic condition for the characteristics methods. Error estimate is derived when the imposed external flows are smoothly distributed. Numerical experiments in three dimensions are given.