cBoundary Condition

In this paper, we introduce the Navier-Stokes equations with a new boundary condition. In this context, we show the existence and uniqueness of the solution of the weak formulation associated with the proposed problem. To solve this latter, we use the discretization by mixed finite element method. In addition, two types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true error. In order to evaluate the performance of the method, the numerical results are compared with some previously published works and with others coming from commercial code like ADINA system.


I. INTRODUCTION
This paper describes a numerical solutions of Navier-stoks equations with a new boundary condition generalizes the will known basis conditions, especially the Dirichlet and the Neumann conditions.So, we prove that the weak formulation of the proposed modelling has an unique solution.To calculate this latter, we use the discretization by mixed finite element method.Moreover, we propose two types of a posteriori error indicator which are shown to give global error estimates that are equivalent to the true error.To compare our solution with the some previously ones, as ADINA system, some numerical results are shown.This method is structured as a standalone package for studying discretization algorithms for PDEs and for exploring and developing algorithms in numerical linear and nonlinear algebra for solving the associated discrete systems.It can also be used as a pedagogical tool for studying these issues, or more elementary ones such as the properties of Krylov subspace iterative methods [15].The latter two PDEs constitute the basis for computational modeling of the flow of an incompressible Newtonian fluid.For the equations, we offer a choice of two-dimensional domains on which the problem can be posed, along with boundary conditions and other aspects of the problem, and a choice of finite element discretizations on a quadrilateral element mesh.
Whereas the discrete Navier-Stokes equations require a method such as the generalized minimum residual method (GMRES), which is designed for non symmetric systems [15].
The key for fast solution lies in the choice of effective preconditioning strategies.The package offers a range of options, including algebraic methods such as incomplete LU factorizations, as well as more sophisticated and state-of-theart multigrid methods designed to take advantage of the structure of the discrete linearized Navier-Stokes equations.In addition, there is a choice of iterative strategies, Picard iteration or Newton's method, for solving the nonlinear algebraic systems arising from the latter problem.
A posteriori error analysis in problems related to fluid dynamics is a subject that has received a lot of attention during the last decades.In the conforming case there are several ways to define error estimators by using the residual equation.in particular, for the Stokes problem, M. Ainsworth, J. Oden [10], C.Carstensen, S.A. Funken [12], D.Kay, D.Silvester [13] and R.Verfurth [14], introduced several error estimators and provided that that they are equivalent to the energy norm of the errors.Other works for the stationary Navier-Stokes problem have been introduced in [5,8,15,16].
The plan of the paper is as follows.Section II presents the model problem used in this paper.The weak formulation is presented in section III.In section IV, we show the existence and uniqueness of the solution.
The discretization by mixed finite elements is described in section V. Section VI introduced two types of a posteriori error bounds of the computed solution.Numerical experiments carried out within the framework of this publication and their comparisons with other results are shown in Section VII.

II. GOVERNING EQUATIONS
We will consider the model of viscous incompressible flow in an idealized, bounded, connected domain in We also assume that  has a polygonal boundary     : , so n  that is the usual outward-pointing normal.www.ijacsa.thesai.org The vector field u  is the velocity of the flow and the scalar variable p represents the pressure.
Our mathematical model is the Navier-stoks system with a new boundary condition ( 3  is the divergence and 2  is the Laplacien operator, ),  There are two strictly positive constants , and Where are and , c b a the function continuous defined on .

III. THE WEAK FORMULATION
We define the following spaces:   (10) The standard weak formulation of the Navier-Stokes flow problem (1) -( 2)-( 3) is the following: . ) , ( . ) , ( And the tri-linear forms The underlying weak formulation (11) may be restated as: In the sequel we can assume that .0    g

IV. THE EXISTENCE AND UNIQUENESS OF THE SOLUTION
In this section we will study the existence and uniqueness of the solution of problem (18), for that we need the following results.
Proof: it is easy.
2) The bilinear form b is satisfies the inf-sup: There exists a constant By Green formula, we have 3) It's easy, just take (28 4) It' suffices to apply (29).5) The same proof of V.Girault and P.A. Raviart in [6] page 115.
According the theorems 1.2 and 1.4, chapter IV in [6], the results (18)-(30) ensure the existence at least one pair We define Then a well-know (sufficient) condition for uniqueness is that forcing function is small in the sense that (it suffices to apply theorems 1.3 and 1.4 chapter IV in [6]).www.ijacsa.thesai.orgTheorem 4.6.Assume that ν and Then there exists an unique   Proof.The some proof of theorem 2.4 chapter IV in [6].

V. MIXED FINITE ELEMENT APPROXIMATION
In this section we assume that


and T N the set of its edges and vertices, respectively.
We let denotes the set of all edges split into interior and boundary edges.
We denote by T h the diameter of a simplex, by T h the diameter of a face E of T, and we set A discrete weak formulation is defined using finite The discrete version of ( 15) is: ,  ,  , : such that and We define the appropriate bases for the finite element spaces, leading to non linear system of algebraic equations.Linearization of this system using Newton iteration gives the finite dimensional System: : such that and are the non linear residuals associated with the discrete formulations (36).To define the corresponding linear algebra problem, we use a set of vectorvalued basis functions   We introduce a set of pressure basis functions   Where u n and p n are the numbers of velocity and pressure basis functions, respectively.We find that the discrete formulation (37) can be expressed as a system of linear equations .0 0 The system is referred to as the discrete Newton problem.The matrix 0 A is the vector Laplacian matrix and 0 B is the divergence matrix The vector-convection matrix N and the Newton derivative matrix W are given by ) .

VI. A POSTERIORI ERROR ESTIMATOR
In this section we propose two types of a posteriori error indicator, a residual error estimator and local Poisson problem www.ijacsa.thesai.orgestimator, which are shown to give global error estimates that are equivalent to the true error.

A. A Residual Error Estimator
The bubble functions on the reference element For a boundary edge  which is only defined on the edge E also denotes its natural extension to the element T.
From the inequalities (50) and (51), we established the following lemma: Lemma 6.2.Let T be a rectangle and . , in the other three edges of rectangle T, it can be extended to the whole of Ω by setting Using the inequalities (19), ( 50) and (51) gives We recall some quasi-interpolation estimates in the following lemma.

Lemma 6.3. Clement interpolation estimate: Given
), by averaging as in [4].For any We let   p u,  denote the solution of (18) and let denote the solution of (36) with an approximation on a rectangular subdivision .
h T Our aim is to estimate the velocity and the pressure errors (55) and the components in (55) are given by With the key contribution coming from the stress jump associated with an edge E adjoining elements T and S: The global residual error estimator is given by: ;

Our aim is bound
With these two functions we have the following lemmas: Lemma 6.4.For any h T T  we have:


By applying the Green formula and

Proof.
i) The same proof of (56).
We define also the following functional Proof.Using (27), (32) and (35), we have that for Using this result and (60), we obtain ), ( and all for , , , Next, we establish the equivalence between the norms of The same proof of theorem 3 in [9].Theorem 6.9. We gather (66) and (67) to get For any mixed finite element approximation (not necessarily inf-sup stable) defined on rectangular grids h T , the residual estimator R  satisfies: be the clement interpolation of .v  Using (63), ( 61) and (62), give www.ijacsa.thesai.org52) and (53), then gives Finally, using (65) gives: According to theorem 6.8, we have This establishes the upper bound.
Turning to the local lower bound.First, for the element residual part, we have: , using (56) and (57) gives: In addition, from the inverse inequality (47) , .

B. The Local Poisson Problem Estimator.
The local Poisson problem estimator defined as: . any for In addition, from the inverse inequalities (47), And using (77), to get Next, we let 78), ( 50) and ( 51 , From this result and the inverse inequalities (49), give We have also VII.NUMERICAL SIMULATION In this section some numerical results of calculations with mixed finite element Method and ADINA system will be presented.Using our solver, we run the flow over an obstacle [15] with a number of different model parameters.Equally distributed streamline plot associated with a 32×80 square grid The two solutions are therefore essentially identical.This is very good indication that my solver is implemented correctly.Pressure plot for the flow with a 32 × 80 square grid.www.ijacsa.thesai.orgNumerical results are presented to see the performance of the method, and seem to be interesting by comparing them with other recent results.
called the kinematic viscosity,  is the gradient, .

Theorem 4 . 1 . 2 c
There are two strictly positive constants 1 c and such that:


is of rectangles sharing at least one edge with element T, T  ~is the set of rectangles sharing at least one vertex with T. Also, for an element edge E, E  denotes the union of rectangles sharing E, while E  ~is the set of rectangles sharing at least one vertex whit E.Next, T  is the set of the four edges of T we denote by ) (T

F
the affine map form T. to T For an interior edge 

Lemma 6 . 1 .
With these bubble functions, ceruse et al ([3], lemma 4.1] established the following lemma.Let T be an arbitrary rectangle in and

C
are tow constants which only depend on the element aspect ratio and the polynomial degrees 0 with generic constants c and C In addition,E v 

Theorem 6 . 8 .
Let the conditions of theorem 4.6 hold.There exist two positive constants , and 2 1


Note that the constant C in the local lower bound is independent of the domain, and .

Consequence 6 . 12 .
80) Combining (78), (79) and (80), establishes the upper bound in the equivalence relation.For the lower, we need to use (65): www.ijacsa.thesai.orgv  is zero at the four vertices of T, a scaling argument and the usual trace theorem, see e.g.[15, Lemma 1.5], shows that v  inequalities with (82) immediately gives the lower bound in the equivalence relation.For any mixed finite element approximation (not necessarily inf-sup stable) defined on rectangular grids h T , the residual estimator P  satisfies: Note that the constant C in the local lower bound independent of the domain.

Example:
Flow over an obstacle.This is another classical problem.The domain is  and is associated with modelling flow in a rectangular channel with a square cylindrical obstruction.A Poiseuille profile is imposed on the Inflow boundary noflow (zero velocity) condition is imposed on the obstruction and the top and bottom walls.A Neumann condition is applied at the outflow boundary which automatically sets the mean outflow pressure to zero. a disconnected rectangular region

TABLE II .
A residual error estimator for Flow over an obstacle with Reynolds number Re = 1000.We were interested in this work in the numeric solution for two dimensional partial differential equations modelling (or arising from) model steady incompressible fluid flow.It includes algorithms for discretization by mixed finite element methods and a posteriori error estimation of the computed solutions.Our results agree with Adina system.