A New Finite Element Method for Darcy-Stokes-Brinkman Equations

We present a new finite element method for Darcy-Stokes-Brinkman equations using primal and dual meshes for the velocity and the pressure, respectively. Using an orthogonal basis for the discrete space for the pressure, we use an efficiently computable stabilization to obtain a uniform convergence of the finite element approximation for both limiting cases. 1. Introduction Recently developing an efficient finite element method for Darcy-Stokes-Brinkman problem has become an active area of research [1–5]. This mathematical model is often called a generalized Stokes problem [6] and is also a special case of Oseen's equations when the convective term is absent [1]. This model incorporates both Darcy and Stokes problem. If the viscosity vanishes, the model becomes the Darcy problem, and if another parameter is set to zero, it reduces to Stokes problem. Therefore, this is important when studying the coupling between Stokes and Darcy equations. Some applications of Darcy-Stokes-Brinkman problem are given in [4]. This model also arises from the time discretization of time-dependent Stokes equations. In this paper we propose a new finite element method for this problem using stabilization based on a local projection. Our technique is the combination of the local projection stabilization for Stokes equations [7] and the grad-div stabilization proposed earlier for the Stokes equations [8]. We note that we mainly focus on the Darcy-Stokes-Brinkman model so that the divergence of the velocity field is not identically zero but is given as a function in -space as in [3, 4]. Using a standard continuous linear, bilinear, or trilinear finite element space for the velocity, the grad-div stabilization [8] is to be modified in this case. Since the modification involves projecting the divergence of the velocity field onto the finite element space for the pressure, we first use a discontinuous pressure space in the finite element scheme. This leads to an efficient realization of the stabilization. The finite element space for the pressure is the piecewise constant function space on the dual mesh. The inf-sup condition is satisfied using an element-wise defined bubble function in each element. This paper is organized as follows. We present the boundary value problem for the Darcy-Stokes-Brinkman problem in the next section. We present our finite element method, propose stabilization, and analyze the discrete problem in Section 3. We prove an optimal a priori error estimate for both cases in this section. Finally, a conclusion is drawn in the last section. 2.