A two-grid Eulerian–Lagrangian localized adjoint method to miscible displacement problems with dispersion term

Abstract In this paper, we propose a novel two-grid method of Eulerian–Lagrangian localized adjoint method (ELLAM)-mixed finite element method (MFEM) solutions for nonlinear and coupled miscible displacement problems, which describe complex fluid flow processes in porous media. An ELLAM is used to solve the nonlinear convection–diffusion equation for concentration, and we present a MFEM to compute the pressure equation for the pressure and Darcy velocity. A two-grid scheme is investigated to linearize and decouple the mixed-method equations. It is illustrated that the coarse space can be selected as coarse as H = O ( h 1 ∕ 2 ) and the asymptotically optimal approximation as the nonlinear schemes can be still achieved. As a result, a large class of such nonlinear and coupled problems can be solved efficiently with large time steps.