Mixed methods with dynamic finite-element spaces for miscible displacement in porous media

AbstractMiscible displacement in porous media is modeled by a nonlinear coupled system of two partial differential equations. We approximate the pressure equation, which is elliptic, and the concentration equation, which is parabolic but normally convection-dominated, by the mixed methods with dynamic finite-element spaces, i.e., different number of elements and different basis functions are adopted at different time levels; and the approximate concentration is projected onto the next finite-element space in weighted L2-norm for starting a new time step. This allows us to make local grid refinements or unrefinements and basis function improvements. Two fully discrete schemes are presented and analysed. Error estimates show that these methods have optimal convergent rate in some sense. The efficiency and capability of the dynamic finite-element method are commented for accurately solving time-dependent problems with localized phenomena, such as fronts, shocks, and boundary layers