Optimal error analysis of Crank-Nicolson lowest-order Galerkin-mixed finite element method for incompressible miscible flow in porous media

Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest-order Galerkin-mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest order Raviart-Thomas mixed element pair is used for the Darcy velocity and pressure. The existing error estimate of the method inL(2)-norm is in the orderO hp+hc2in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial meshes, excluding the most commonly used meshh = h(p) = h(c). This paper focuses on new and optimal error estimates of a linearized Crank-Nicolson lowest-order Galerkin-mixed finite element method (FEM), where the second-order accuracy for the concentration in both time and spatial directions is established unconditionally. The key to our optimal error analysis is an elliptic quasi-projection. Moreover, we propose a simple one-step recovery technique to obtain a new numerical Darcy velocity and pressure of second-order accuracy. Numerical results for both two and three-dimensional models are provided to confirm our theoretical analysis.