Abstract
In this paper, we develop an a-priori error analysis of a new unified mixed finite element method for the coupling of fluid flow with porous media flow in RN, N ∈ {2,3}, on isotropic meshes. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The approach utilizes a modification of the Darcy problem which allows us to apply a variant nonconforming Crouzeix-Raviart finite element to the whole coupled Stokes-Darcy problem. The well-posedness of the finite element scheme and its convergence analysis are derived. Finally, the numerical experiments are presented, which confirm the excellent stability and accuracy of our method.
Highlights
There are many serious problems currently facing the world in which the coupling between groundwater and surface water is important
The approach utilizes a modification of the Darcy problem which allows us to apply a variant nonconforming Crouzeix-Raviart finite element to the whole coupled Stokes-Darcy problem
In [22], a conforming unified finite element has been proposed for the modified coupled Stokes-Darcy problem in a plane domain, which has simple and straightforward implementations
Summary
There are many serious problems currently facing the world in which the coupling between groundwater and surface water is important These include questions such as predicting how pollution discharges into streams, lakes, and rivers its way into the water supply. In [22], a conforming unified finite element has been proposed for the modified coupled Stokes-Darcy problem in a plane domain, which has simple and straightforward implementations. We use a variant nonconforming Crouzeix-Raviart finite element method that has so many advantages for the velocities and piecewise constant for the pressures in both the Stokes and Darcy regions, and apply a stabilization term penalizing the jumps over the element edges of the piecewise continuous velocities.
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