F. Fairag, Numerical computations of viscous, incompressible flow problems using a two-level finite element method. SIAM J. Sci. Comput. 24, No.6, 1919-1929 (2003). Preprint: [pdf] [link] Abstract. We consider two-level finite element discretization methods for the stream function formulation of the Navier--Stokes equations. The two-level 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 two-level method can be implemented to approximate efficiently solutions to the Navier--Stokes 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.

  F. Fairag, Two-level finite element technique for pressure recovery from stream function formulation of the Navier-Stokes equations. Barth, Timothy J. (ed.) et al., Multiscale and multiresolution methods. Theory and applications. Berlin: Springer. Lect. Notes Comput. Sci. Eng. 20, 297-306 (2002). [ISBN 3-540-42420-2] Preprint: [pdf]  [ps] Summary: We consider two-level finite element discretization for the stream function formulation of Navier-Stokes equations. The two-level 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 two-dimensional driven cavity flow is considered. Stream function contours are displayed, showing the main features of the flow.

  F. Fairag, A two-level finite-element discretization of the stream function form of the Navier-Stokes equations. Comput. Math. Appl. 36, No.2, 117-127 (1998). Preprint: [pdf] Summary: We analyze a two-level method of discretizing the stream function form of the Navier-Stokes equations. This report presents the two-level algorithm and error analysis for the case of conforming elements. The two-level 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 Clough-Tocher triangles or the Bogner-Fox-Schmit rectangles are used, then the coarse and fine meshes are related by $h= O(H^{3/2}|\ln H|^{1/4})$.

  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). Preprint: [pdf]  [ps] 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 velocity-pressure 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 Navier-Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.

  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), 287-305 (2008) Preprint: [pdf]  [ps] 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 Ladyzhenskaya-type. 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 Navier-Stokes 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.

  F. Fairag. Finite Difference Method on Triangulation. Submitted to SIAM Journal on Scientific Computing. (2005). Preprint: [pdf]  [ps] 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 non-nodal points. Then, the selected triangular element is used to approximate the function values of the non-nodal 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.

  F. Fairag. Finite element computations of pure-streamfunction equation of the Ladyzhenskaya equations for incompressible fluid. In Proceeding of 8th WSEAS International Conference on Applied Mathematics. Spain, 2005. Preprint: [pdf] Abstract. In this paper we consider a pure-streamfunction equation of Ladyzhenskaya equations. For certain values of the parameters of the equation, the studied equation becomes identical to the pure-streamfunction equation of the Navier-Stokes equations. A weak form, a finite element method approximation procedures and an iterative method for solving the discrete nonlinear problems are provided. Using the Bogner-Fox-Schmidt element, the steady 2-D 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.

  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 two-level finite element discretization methods for the stream function formulation of the Navier--Stokes equations. The two-level 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 two-level method can be implemented to approximate efficiently solutions to the Navier--Stokes 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.

  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 two-level finite element discretization methods for the stream function formulation of the Navier--Stokes equations. The two-level 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 two-level method can be implemented to approximate efficiently solutions to the Navier--Stokes 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.

  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 two-level finite element discretization methods for the stream function formulation of the Navier--Stokes equations. The two-level 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 two-level method can be implemented to approximate efficiently solutions to the Navier--Stokes 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.

  F. FAIRAG, Finite Element Computations for Viscous Incompressible Flows Using Pure-Streamfunction Equation of the Ladyzhenskaya Equations. WSEAS Trans. Math. 4 (2005), no. 4, 493--499. 65N30 (76M10) Preprint: [pdf] Abstract.  The pure-streamfunction equation of Ladyzhenskaya equations is considered which, for specific parameters, is identical to the pure-streamfunction equation of the Navier-Stokes 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 2-D incompressible flow in a driven cavity using the Bogner-Fox-Schmidt 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 Navier-Stokes equations.

  F. FAIRAG, A Two-Level Discretization Method For The Streamfunction Form Of The Navier-Stokes Equations. Ph.D. Dissertation, Math. Dept., University of Pittsburgh, Pittsburgh, PA, USA, (1998). [under supervision of Prof. William Layton] Preprint: [pdf] Abstract. We analyze a two-level finite element method for the streamfunction formulation of the Navier-Stokes equations. This report presents the two-level 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 two-level 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 Navier-Stokes 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 two-level algorithm is derived which can be used as an indicator for an assessment of the reliability of the results.

  Mohamed A. El-Gebeily, 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.