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

Abstract

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.