Mixed finite element method and the characteristics-mixed finite element method for a slightly compressible miscible displacement problem in porous media

• We introduce an approximate scheme for single phase, two-component slightly compressible miscible flow in porous media.
• This scheme is constructed by standard mixed finite element and a characteristic-mixed finite element method.
• Optimal L2-norm error estimates are derived for the concentration and the diffusive flux simultaneously.

Single-phase, two-component slightly compressible miscible flow in porous media, is governed by a system of nonlinear partial differential equations. The density of the mixture is related to the pressure and the concentration. An approximate scheme is introduced for this system, which is constructed by two methods. A standard mixed finite element is used for the pressure equation. A characteristic-mixed finite element method is presented for the concentration equation. Characteristic approximation is applied to handle the convection part of the concentration equation, and a lowest order mixed finite element spatial approximation is adopted to deal with the diffusion part. Thus, the scalar unknown concentration and the diffusive flux can be approximated simultaneously. In order to derive the optimal L2-norm error estimates, a post-processing step is included in the approximation to the scalar unknown concentration. A numerical experiment is presented finally to validate the theoretical analysis.