First-Order System Least-Squares for Darcy-Stokes Flow

The subject of this paper is a first-order system least-squares formulation for the Stokes equation which remains uniformly valid in the limit of vanishing viscosity. For this so-called Darcy-Stokes flow problem we establish continuity and coercivity of the corresponding least-squares functional in appropriate norms. Two types of finite element spaces for the approximation of the velocity field are investigated in detail: the well-known Raviart-Thomas elements and an element recently introduced by Mardal, Tai and Winther specifically for mixed approaches to Darcy-Stokes flow. The computational results derived with next-to-lowest order Raviart-Thomas elements as well as the Mardal-Tai-Winther elements confirm the analysis.