Mixed Finite Element-Characteristic Mixed Finite Element Method for Simulating Three-Dimensional Incompressible Miscible Displacement Problems

A mixed finite element with the characteristics is presented as a local conservative numerical approximation for an incompressible miscible problem in porous media. A mixed finite element (MFE) is used for the pressure and Darcy velocity, and a characteristic method is for the saturation. The convection term is discretized along the characteristic direction and the diffusion term is discretized by zero-order mixed finite element method. The method of characteristics confirms the strong stability without numerical dispersion at sharp fronts. Moreover, large time step is possibly adopted without any accuracy loss. The scalar unknown function and the adjoint vector function are obtained simultaneously and the law of mass conservation holds in every element by the zero-order mixed finite element discretization of diffusion flux. In order to derive the optimal $3/2$-order error estimate in $L^2$ norm, a post-processing technique is included in the approximation to the scalar unknown saturation. This method can be used to solve the complicated problem.