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

First-Order System Least-Squares for Darcy-Stokes Flow,10.1137/050638163,Siam Journal on Numerical Analysis,Garvin Danisch,Gerhard Starke

First-Order System Least-Squares for Darcy-Stokes Flow   (Citations: 3)
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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. 1. Introduction. Our purpose in this paper is to present a least-squares finite element method for Darcy-Stokes flow which remains valid for arbitrarily small vis- cosity. This type of singular perturbation problem was studied before in (11) where a successful mixed finite element approach is presented. The mixed variational formula- tion of (11) is of saddle point structure with its well-known limitation on the admissible combinations of finite element spaces. One of the motivations for the development of the least-squares approach presented in this paper is the greater flexibility in the choice of finite element spaces which is not restricted by a compatibility condition. In the limit of vanishing viscosity, our least-squares formulation turns into the one proposed in (8). The approach in (8) constructs approximations for the pressure and the velocity in H1() and H(div,), respectively. In the viscous case, however, an approximation for the velocity is sought in H1() 2 instead. This is achieved by an augmentation with a least-squares functional along the edges of the triangulation over the jump of the tangential component and by introducing the velocity gradient as an additional variable. The case of small viscosity is handled by an appropriate weighting of the components in the least-squares functional. Our main motivation for this work comes from the treatment of shallow water systems treated with the method of characteristics for time discretization. In this context, linearization of the boundary value problems at each time-step leads to flow problems of Darcy-Stokes type. Shallow water flow is described by the scalar water level and by the velocity field. These process variables are directly approximated by the first-order system least-squares formulation treated in this paper. The exten- sion to shallow water systems including a viscosity term is therefore straightforward. For vanishing viscosity that approach reduces to the first-order system least-squares method investigated in (14). Among the most popular methods for the case µ = 0 is the Raviart-Thomas mixed finite element method which couples, for example, lowest-order Raviart-Thomas elements for the flux with piecewise constant functions for the scalar variable. This approach is well studied in the case of the linear first-order Darcy-type system (see e.g. (4, Section III.5)) as well as for the shallow water system without viscosity (see
Journal: Siam Journal on Numerical Analysis - SIAM J NUMER ANAL , vol. 45, no. 2, pp. 731-745, 2007
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