A stabilized finite element method based on two local Gauss integrations for a coupled Stokes–Darcy problem

In this paper, a stabilized mixed finite element method for a coupled steady Stokes–Darcy problem is proposed and investigated. This method is based on two local Gauss integrals for the Stokes equations. Its originality is to use a difference between a consistent mass matrix and an under-integrated mass matrix for the pressure variable of the coupled Stokes–Darcy problem by using the lowest equal-order finite element triples. This new method has several attractive computational features: parameter free, flexible, and altering the difficulties inherited in the original equations. Stability and error estimates of optimal order are obtained by using the lowest equal-order finite element triples (P1−P1−P1)(P1−P1−P1) and (Q1−Q1−Q1)(Q1−Q1−Q1) for approximations of the velocity, pressure, and hydraulic head. Finally, a series of numerical experiments are given to show that this method has good stability and accuracy for the coupled problem with the Beavers–Joseph–Saffman–Jones and Beavers–Joseph interface conditions.