A Finite Element Approximation of the Stokes Equations

In this work, a numerical solution of the incompressible Stokes equations is proposed. The method suggested is based on an algorithm of discretization by the unstable of Q 1 – P 0 velocity/pressure finite element approximation. It is shown that the inf-sup stability constant is O(h) in two dimensions and O() in three dimensions. The basic tool in the analysis is the method of modified equations which is applied to finite difference representations of the underlying finite element equations. In order to evaluate the performance of the method , the numerical results are compared with some previously published works or with others coming from commercial code like Adina system.


Introduction
It is universally recognized that discretization schemes for Stokes and Navier-Stokes equations are subject to an inf-sup or div-stability condition [1].The stability requirement is manifested in practical computations by the predominance of staggered grid finite volume discretizations, and the existence of unnatural velocity-pressure finite element combinations.These typically involve velocity bubble functions, or else have a macro-element definition of the velocity field.
A number of stabilization methods for inf-sup unstable approximations have been developed during the last three decades.These methods can be classified into two kinds.The first one is residual based stabilization, e.g. the absolutely stabilized method introduced by Douglas and Wang [2] and the Galerkin least square methods introduced by Franca and Hughes [3].The other kind consists of pressure stabilized methods, and includes the global pressure jump stabilized method of Hughes and Franca [4].
In this paper the low order conforming finite element methods like 1 0 (trilinear/bilinear velocity with constant pressure) for incompressible flow problems are characterised.

Q P 
The 1 0 finite element method is particularly controversial [1].Despite being damned by theoreticians after the discovery of "weakly singular" modes [5,6], 1 0 is widely used in practice.For the first time a clean analysis of the instability mechanism is presented.

Q P 
Section 2 presents the model problem used in this paper.The discretization by finite elements is described in Section 3. Numerical experiments carried out within the framework of this publication and their comparisons with other results are shown in Section 4.

Governing Equations
We consider the Stokes equations for the flow of an incompressible viscous fluid where denotes the unit square with boundary v is a given constant called the kinematic viscosity, u is the fluid velocity, p is the pressure field and  is the gradient operator.
The geometry of the domain and boundary condition are shown in Figure 1.

Finite Element Approximation
In this section we introduce the finite element approximation of the Stokes equations in two dimensions, and we proceed to describe a means of estimating the relevant constant in the inf-sup condition.An alternative formulation is discussed at the end of this section.Assuming a grid of square elements K, so that h = 1/n, an approximation based on the element uses function spaces for velocity and pressure, respectively, where   1 Q K is the space of bilinear functions and is the space of constant functions on K.The nodal positions of the 1 0 element are illustrated in Figure 2. Denoting the velocity space satisfying homogeneous boundary conditions by , the finite element formulation is: which interpolate the data h  u g at boundary nodes and such that with standard interpolating bases for the spaces h X and , this leads to the matrix system h W 0 0 0 where U, contain the nodal values of the x and y components of the approximation to h at the internal vertices, and The satisfaction of the inf-sup condition is dependent on the eigenvalues where is the 2 N h I  p p mass matrix corresponding to the pressure space [7], the discrete divergence operator and K is the vector Laplacian 0 0 Since the discrete velocity field is specified everywhere on the boundary, the discrete Stokes operator has a two-dimensional null space spanned by the hydrostatic and chequerboard pressure modes [8].Consequently, (6) has only 2 2 n  non-zero eigenvalues and we can order the eigenvalues so that It may also be shown [7], that the nonzero eigenvalues of the "dual" problem coincide with those of (6).We shall find it more convenient to work with the related eigenproblem 0 0 0 0 0 0 0 which reduces to (6) on elimination of U and V and to (9) on elimination of P.This system has an eigenvalue 1   of multiplicity equal to the dimension of the null space of the discrete divergence operator, that is , and the remaining Applying the analysis of Brezzi and Fortin [1] or Malkus [7] the inf-sup stability of the system (4) (or, equivalently, (5)) is determined by the square root of the smallest non zero eigenvalue of (6) that is and we shall, accordingly, refer to 3   as the critical eigenvalue of (10) and denote it by  

3
. Experimental evidence reported in [2] suggests that 0 sue fu vestigate this is rther we write the constituent equations of (10) in finite difference form and then appeal to the method of modified equations.This will allow us to establish the precise behavior of 3  for small values of h.The x and y components of ocity at the grid point vel (lh, mh), l, , , . The system of Equations (10) (each divided by may then be expressed as , 2 are the usual central difference and averaging operators, respectively, and 2 h  denotes the discrete Laplacian generated by bilinear elements It then follows that    , , , , , it is easy to see that ( 24) is the Q 1 -P 0 discretization of the modified system (25) In the limit we obtain the reduced problem 0 in , pendent variables For each of the de , , . This, being a second order elliptic eigenvalue only one boundary condition whereas co efore repre problem, requires the system (26) ntains two.The Equations (25) ther sent a singularly perturbed system [9] whose solutions will, in general, contain boundary layers.Bearing this in mind, we may identify the smallest non zero eigenvalue of (26) as associated with which there are two eigenfunctions, having outer expansions It is important to note that since 1 0 u  on  , ve it exrtical hibits boundary layers of width O(h) along the boundaries x = 0, 1.Similarly, 2  has boundar layers of ounda y = width O(h) along the horizontal b ries y 0, 1.We also note that, for this modified system, a "pressure gradient" in the y-direction induces a "flow" in the xdirection and vice-versa.We summarise in the form of a theorem.
Theorem 1: The critical eigenvalue of ( 10) is given by and the inf-sup constant for Q 1 -P 0 elements is given by The eigenspace co  rresponding to *  has dimension two and is spanned by mutually orth given, approximately, by ogonal eigenvectors and st strongly in the instabilities manifest themselves mo the pressure field.This will be explored further in the next section in connection with the model lid driven cavity problem.
An alternative Stokes formulation to (1) is the stress divergence form [1]: u g which after discretisation gives a matrix system which is slightly different to (5) above.If we discreti Q 1 -P 0 and follow the procedure described se (33) using by (17) for changing variables, then it is easily shown that the resulting reduced problem is also of the form (26) except that the factor 8 3 is replaced by 4. In this case we have the result stated below.Theorem 2: The inf-sup constant for Q 1 -P 0 approximation of (33 s given by ) i e has dimension two and is spanned by mutually orthogonal eigenvectors given, approximately, by (31) and (32).

4.
results of calculations ith finite element method and ADINA system will be
presented.Using our solver, we p We consider the critical eigenvalue of Theorem 1.The numerically computed eigenvalues j  are presented in Table 1.The smallest eigenvalues are clearly tending to zero like O(h 2 ).The largest eigenvalue is converging to an asymptotic limit of unity.Note also that for "unstable" eigenvalues: and 1

Conclusions
We were interested in this work in the numerical solution for two dimensional differential equations model steady incompressible flow.It includes algorithms for discretization by finite element methods.For the test of drivencavity flow, the particles in the body of the fluid move in a circular trajectory.
Our results agree with Adina System.Numerical results are presented to see the performance of the method, and seem to be interesting by comparing them with other recent results.

Acknowledgments
the referee for his/her helpful suggestions.
The authors would like to express their sincere thanks for

Figure 1 .
Figure 1.Geometry of the domain and boundary condition for the velocity .u Figure 2.  1 0 Q P element ( • two velocity components; O

Figure 2
Figure 2 shows the element and Figure 3 sh ith (13) we have the boundary conditions

Figure 3 .
Figure 3. Pressure and the components of the velocity at e grid point.th

2 O
) at internal nodes of the domain and boundary nodes.The amplitudes of d pressure contributions to th r h , which explains why , in computations,

Figures 4 6 an
and 5 show, respectively, the contours of pressure component of the first and the second u eigenvectors corresponding to nstable   , whereas Figures the u-velocity com onent along the p lin nd the v-velocity component along th vertical center e a e horizontal center line are shown in Figure 8.In this figure, we have the excellent agreement between the computed results and the results computed with ADINA system.

Figure 4 .
Figure 4. Contours of pressure component of the first unstable eigenvector corresponding to   .

Figure 5 .
Figure 5. Contours of pressure component of the second unstable eigenvector corresponding to   .

Figure 6 .
Figure 6.Contours of pressure component of the unstable eigenvector corresponding to 5  .

Figure 8 .
Figure 8.The velocity component u at vertical center line (a), and the velocity component v at horizontal center line (b) with a 129 × 129 grid.