Details of Research Outputs

The time discrete scheme of characteristics type is especially effective for convection-dominated diffusion problems. The scheme has been used in various engineering areas with different approximations in spatial direction. The lowest-order mixed method is the most popular one for miscible flow in porous media. The method is based on a linear Lagrange approximation to the concentration and the zero-order Raviart--Thomas approximation to the pressure/velocity. However, the analysis for the lowest-order characteristics-mixed finite element method (FEM) has not been well done, although significant effort has been made in the last several decades. In all previous works, only first-order accuracy in spatial direction was proved under certain time-step and mesh size restrictions. The main purpose of this paper is to establish optimal error estimates, i.e., the second order in the $L^2$-norm for the concentration and the first order for the pressure/velocity, while the concentration is the more important physical component. For this purpose, an elliptic quasi-projection is introduced in our analysis to clean up the pollution of the numerical velocity through the nonlinear dispersion-diffusion tensor and the concentration-dependent viscosity. Moreover, the numerical pressure/velocity of the second-order accuracy can be obtained by re-solving the (elliptic) pressure equation at a given time level with a higher-order approximation. Numerical results presented in this paper confirm our theoretical analysis.