A fully mixed virtual element method for Darcy–Forchheimer miscible displacement of incompressible fluids appearing in porous media

Abstract

Abstract The incompressible miscible displacement of two-dimensional Darcy–Forchheimer flow is discussed in this paper, and the mathematical model is formulated by two partial differential equations, a Darcy–Forchheimer flow equation for the pressure and a convection–diffusion equation for the concentration. The model is discretized using a fully mixed virtual element method (VEM), which employs mixed VEMs to solve both the Darcy–Forchheimer flow and concentration equations by introducing an auxiliary flux variable to ensure full mass conservation. By using fixed point theory, we proved the stability, existence and uniqueness of the associated mixed VEM solution under smallness data assumption. Furthermore, we obtain optimal error estimates for concentration and auxiliary flux variables in the $\texttt {L}^{2}$- and $\textbf {L}^{2}$-norms, as well as for the velocity in the $\textbf {L}^{2}$-norm. Finally, several numerical experiments are presented to support the theoretical analysis and to illustrate the applicability for solving actual problems.