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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.

Recently developing an efficient finite element method for Darcy-Stokes-Brinkman problem has become an active area of research [

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 [

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

This section is devoted to the introduction of the boundary value problem of the Darcy-Stokes-Brinkman flow. Let

Here we use standard notations

The weak formulation of the Darcy-Stokes-Brinkman problem is to find

We note that, when the viscosity

With this change of the character of the solution, when the viscosity

Therefore now we use the space

We consider a quasiuniform triangulation

For nonnegative integer

A typical element

The finite element space of displacement is taken to be the space of continuous functions whose restrictions to an element

Let the set of all vertices in

A dual mesh

Primal and dual meshes for a triangular grid.

In the following we use a generic constant

We call the control volume mesh

Now

For the reference element

Then defining the space of bubble functions

With this definition of the spaces

There exists a constant

We now consider the limiting case when

When

Assuming that

Since this condition is not satisfied for the space

Due to the fact that the basis functions for the space

If the source term

We now show that the bilinear form

One has

Note that

Clearly, we see that the bilinear forms

The discrete saddle point problem (

The error estimate is given in the following theorem.

Let

We have presented a very efficient finite element technique for approximating the solution of Darcy-Stokes-Brinkman equations. Using local projection stabilization for the limiting case of vanishing viscosity we have shown that the error estimate is uniform for both limiting cases.