Abstract: In this paper a mixed finite element-characteristic mixed finite element method is discussed to simulate an incompressible miscible Darcy-Forchheimer problem. The flow equation is solved by a mixed finite element and the approximation accuracy of Darch-Forchheimer velocity is improved one order. The concentration equation is solved by the method of mixed finite element, where the convection is discretized along the characteristic direction and the diffusion is discretized by the zero-order mixed finite element method. The characteristics can confirm strong stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation. In actual computations the characteristics adopts a large time step without any loss of accuracy. The scalar unknowns and its 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 unknowns. Numerical experiments are illustrated finally to validate theoretical analysis and efficiency. This method can be used to solve such an important problem.

Key words: Darcy-Forchheimer miscible displacement, mixed element-characteristic mixed element-postprocessing scheme, local conservation of mass, $3/2$-order error estimates in $L^2$-norm, numerical computation