

[1] F. Fairag, Numerical computations of viscous, incompressible flow problems using a twolevel finite element method. SIAM J. Sci. Comput. 24, No.6, 19191929 (2003). 




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Abstract. We consider twolevel finite element discretization methods for the stream function
formulation of the NavierStokes equations. The
twolevel method consists of solving a small nonlinear system on the coarse
mesh and then solving a linear system on the fine mesh. The basic result
states that the errors between the coarse and fine meshes are related superlinearly. This paper demonstrates that the twolevel
method can be implemented to approximate efficiently solutions to the NavierStokes equations. Two fluid flow calculations are
considered to test problems which have a known solution and the driven cavity
problem. Stream function contours are displayed showing the main features of
the flow. 




[2] F.
Fairag, Twolevel finite element technique for pressure recovery from
stream function formulation of the NavierStokes
equations. Barth, Timothy J. (ed.) et al., Multiscale and multiresolution
methods. Theory and applications. 




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Summary: We consider twolevel finite element discretization for the stream function formulation of NavierStokes equations. The twolevel method consists of
solving a small nonlinear system on coarse mesh, and then solving a linear
system on fine mesh. It is known that the errors between the coarse and fine
meshes are related superlinearly. This paper
presents an algorithm for pressure recovery, and a general analysis of
convergence for the algorithm. The numerical example for twodimensional
driven cavity flow is considered. Stream function contours are displayed,
showing the main features of the flow. 




[3] F. Fairag, A twolevel finiteelement discretization of the stream function form of the NavierStokes equations. Comput. Math. Appl. 36, No.2, 117127 (1998). 




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[pdf] 




Summary: We analyze a twolevel method of discretizing the stream function form of the NavierStokes equations. This report presents the
twolevel algorithm and error analysis for the case of conforming elements.
The twolevel algorithm consists of solving a small nonlinear system on the
coarse mesh, then solving a linear system on the fine mesh. The basic result
states that the error between the coarse and fine meshes are related superlinearly via $\psi
\psi^h_2\le C\left\{ \inf_{w^h\in X^h}\psi w^h_2+ \ln h^{1/2}\cdot\psi \psi^H_1\right\}$. As an example, if the
CloughTocher triangles or the BognerFoxSchmit
rectangles are used, then the coarse and fine meshes are related by $h= O(H^{3/2}\ln H^{1/4})$. 




[4] F. Fairag, Analysis and Finite Elemnet Approximation of A Ladyzhenskaya Model for Viscous Flow in Streamfunction Form. To appear in Journal Of Computational And Applied Mathematics, (2006). 




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Abstract. In this paper we consider a model for the motion of
incompressible viscous flows proposed by Ladyzhenskaya.
The Ladyzhenskaya model is written in terms of the
velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction
equation of the Ladyzhenskaya model and present a
weak formulation and show that this formulation is equivalent to the
velocitypressure formulation. We also present some existence and uniqueness
results for the model. Finite element approximation procedures are presented.
The discrete problem is proposed to be well posed and stable. Some error
estimates are derived. We consider the 2D driven cavity flow problem and
provide graphs which illustrate differences between the approximation
procedure presented here and the approximation for the streamfunction
form of the NavierStokes equations. Streamfunction contours are also displayed showing the
main features of the flow. 




[5] F. Fairag and M. S. Sahimi, The alternating
group explicit (AGE) iterative method for solving a ladyzhenskaya
model for stationary incompressible viscous flow. International
Journal of Computer Mathematics, 85(2), 287305 (2008) 




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Abstract. In this paper, the alternating group explicit (AGE)
iterative method is applied to a nonlinear 4th order PDE describing the flow
of an incompressible fluid. This equation is of a Ladyzhenskayatype.
The AGE method is shown to be extremely powerful and flexible and affords its
users many advantages. Computational results are obtained to demonstrate the
applicability of the method on some problems with known solutions. This paper
demonstrates that the (AGE) method can be implemented to approximate
efficiently solutions to the NavierStokes
equations and the Ladyzhenskaya equations. Problem
with a known solution are considered to test the method and to compare the
computed results with the exact values.
Streamfunction contours and some plots are
displayed showing the main features of the solution. 




[6]
F. Fairag. Finite Difference Method on Triangulation. Submitted to 




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Abstract. In this paper, we present a new method for solving
Partial Differential Equations (PDE). This method combines the use of features
of both Finite Element Methods (FEM) and Finite Difference methods (FDM).
Similar to the FEM, this method uses the triangulation technique and function
approximation. Moreover, it uses direct discretization
of the PDE, similar to the FDM. The basic idea starts by selecting a finite
difference representation of the PDE and a triangular element. The selected
representation involves nodal point and nonnodal points. Then, the selected
triangular element is used to approximate the function values of the
nonnodal points. This method can be seen as a finite difference method when
an irregular nodal arrangement is appropriate for a given problem. Derivation
of the method with remarks is presented as well as several computational
examples with their graphs and tables. 




[7]
F. Fairag. Finite element computations of purestreamfunction
equation of the Ladyzhenskaya equations for
incompressible fluid. In Proceeding of 8^{th} WSEAS
International Conference on Applied Mathematics. 




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[pdf] 




Abstract. In this paper we consider a purestreamfunction equation of Ladyzhenskaya
equations. For certain values of the parameters of the equation, the studied
equation becomes identical to the purestreamfunction
equation of the NavierStokes equations. A weak
form, a finite element method approximation procedures and an iterative method
for solving the discrete nonlinear problems are provided. Using the BognerFoxSchmidt element, the steady 2D incompressible
flow in a driven cavity is solved using a grid mesh of 16X16. Streamfunction contours are also displayed showing the
main features of the flow. 




[8] Mohamed El Shayeb, Faisal Fairag, Zolman bin Hari, Jacqueline Eng Ling Siang, Norhaida bt Ab Razak ,Zulfika Anuar, Utilization of Numerical Techniques to Predict The Thermal Behavior of Wood Column Subjected to Fire Part A: Using Finite Element Methods to Develop Mathematical Model for Wood Column, Proceeding of FEOFS 2005, Bali, Indonesia. 




Preprint: 
[pdf] 




Abstract. We consider twolevel finite element discretization methods for the stream function
formulation of the NavierStokes equations. The
twolevel method consists of solving a small nonlinear system on the coarse
mesh and then solving a linear system on the fine mesh. The basic result
states that the errors between the coarse and fine meshes are related superlinearly. This paper demonstrates that the twolevel
method can be implemented to approximate efficiently solutions to the NavierStokes equations. Two fluid flow calculations are
considered to test problems which have a known solution and the driven cavity
problem. Stream function contours are displayed showing the main features of
the flow. 




[9]
Mohamed El Shayeb, Faisal Fairag, Zolman bin Hari, Jacqueline Eng
Ling Siang, Norhaida bt Ab Razak
,Zulfika Anuar, Utilization
of Numerical Techniques to Predict The Thermal Behavior of Wood Column
Subjected to Fire Part B: Analysis of Column Temperature and Fire Resistance,
Proceeding of FEOFS 2005, Bali, Indonesia. 




Preprint: 
[pdf] 




Abstract. We consider twolevel finite element discretization methods for the stream function
formulation of the NavierStokes equations. The
twolevel method consists of solving a small nonlinear system on the coarse
mesh and then solving a linear system on the fine mesh. The basic result
states that the errors between the coarse and fine meshes are related superlinearly. This paper demonstrates that the twolevel
method can be implemented to approximate efficiently solutions to the NavierStokes equations. Two fluid flow calculations are
considered to test problems which have a known solution and the driven cavity
problem. Stream function contours are displayed showing the main features of
the flow. 




[10] Mohamed El Shayeb, Faisal Fairag, Zolman bin Hari, Jacqueline Eng Ling Siang, Norhaida bt Ab Razak ,Zulfika Anuar, Utilization of Numerical Techniques to Predict The Thermal Behavior of Wood Column Subjected to Fire Part C: Sensitivity Analysis, Proceeding of FEOFS 2005, Bali, Indonesia. 




Preprint: 
[pdf] 




Abstract. We consider twolevel finite element discretization methods for the stream function
formulation of the NavierStokes equations. The
twolevel method consists of solving a small nonlinear system on the coarse
mesh and then solving a linear system on the fine mesh. The basic result
states that the errors between the coarse and fine meshes are related superlinearly. This paper demonstrates that the twolevel
method can be implemented to approximate efficiently solutions to the NavierStokes equations. Two fluid flow calculations are
considered to test problems which have a known solution and the driven cavity
problem. Stream function contours are displayed showing the main features of
the flow. 




[11]
F. FAIRAG, Finite Element Computations for Viscous Incompressible Flows
Using PureStreamfunction Equation of the Ladyzhenskaya Equations. WSEAS Trans. Math. 4 (2005),
no. 4, 493499. 65N30 (76M10) 




Preprint: 
[pdf] 




Abstract. The
purestreamfunction equation of Ladyzhenskaya
equations is considered which, for specific parameters, is identical to the
purestreamfunction equation of the NavierStokes equations. We present a weak form, a
procedure for finite element method approximation and an iterative method for
solving the discrete nonlinear problems. We solve a 2D incompressible flow
in a driven cavity using the BognerFoxSchmidt
element and a grid mesh of 16X16. We display the features of the flow by streamfunction contours and a plot of differences between
the streamfunction of Ladyzhenskaya
and that of the NavierStokes equations. 




[12]
F. FAIRAG, A TwoLevel Discretization Method For
The Streamfunction Form Of The NavierStokes
Equations. Ph.D. Dissertation, Math. Dept., 




Preprint: 
[pdf] 




Abstract. We analyze a twolevel finite element method for
the streamfunction formulation of the NavierStokes equations. This report presents the
twolevel algorithm and a priori error analysis for the case of conforming
elements. The streamfunction formulation in two
dimensions has the great advantages that one solves for only a single scalar
variable rather than a coupled system. Further, the incompressible constraint
is automatically satisfied so there are no compatibility conditions between
velocity and pressure spaces. The disadvantage is that the linear system,
though small, arises from a fourth order problem so it can be very ill
conditioned. The nonlinear system is also, at higher Reynolds numbers, very
sensitive to small perturbations. The twolevel algorithm consists of solving
a small nonlinear system on the coarse mesh, then solving a large linear
system on the fine mesh. The basic results states that the error between the
coarse and fine meshes are related superlinearly. FORTRAN programs for this
algorithm and a complete discussion of these programs are included. These
programs are used to solve the NavierStokes
equations with a known solution in a rectangular domain for a range of
Reynolds numbers to compare one level vs. two level methods in terms of
computer time. Also, we solve the driven flow in a square cavity. These flows
have been widely used as test cases for validating incompressible fluid
dynamics algorithms. Streamfunction contours are
displayed showing the main features of the flow. A posteriori error estimator
for the twolevel algorithm is derived which can be used as an indicator for
an assessment of the reliability of the results. 




[13] Mohamed
A. ElGebeily, Faisal Fairag, Rajai
Alassar, M.B.M. Elgindi,
H^2 solutions for the stream function and vorticity
formulation of the Navier–Stokes equations. To
appear in Applied Mathematics and Computation, (2006) 




Preprint: 
[pdf] 




Abstract. We show that the two dimensional Navier–Stokes equations in the stream function and vorticity form with nonhomogeneous
boundary conditions have a unique solution with a stream function having two
space derivatives. 

