A stabilized mixed finite element method for coupled Stokes and Darcy flows with transport

In this article, we present a stabilized mixed finite element method for solving the coupled Stokes and Darcy flow equations with a solute transport. The mathematical model includes the velocity and pressure equations and concentration equation where the viscosity depends on the concentration. We propose a mixed weak formulation and use the nonconforming piecewise Crouzeix-Raviart finite element, piecewise constant and conforming piecewise linear finite element to approximate velocity, pressure and concentration respectively. The existence, uniqueness of the approximate solution are obtained, and optimal order a priori error estimates are derived. No assumption on the boundness of the infinity norms of approximate velocity or concentration or the restriction about the time-step and spatial meshsize is needed due to a new weak formulation introduced for the concentration equation. Numerical examples are presented to verify the theoretical results.