الفهرس 
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Abstract
The multigrid technique (MG) is one of the most efficient methods for solving
a large class of problems very efficiently. One of these multigrid techniques is the algebraic multi grid (AMG) approach which is developed to solve matrix equations using the principles of usual multigrid methods. In this work, various algebraic
multi grid methods are proposed to solve different problems including: general linear elliptic partial differential equations (PDEs), as anisotropic Poisson equation,
problems with steep boundary layers, as convectional dominant convection-diffusion equations, and nonlinear system of equations as Navier-stokes equations.
In addition, a new technique is introduced for solving convection-diffusion equation by predicting a modified diffusion coefficient (MDC) such that the discretization process applies on the modified equation rather than the original one.
For a class of one-dimensional convection-diffusion equation, we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh
size, then, prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and/or equation coefficients.
Extending the same technique to obtain analytic MDC for other classes of
convection-diffusion equations is not always straight forward especially for higher dimensions. However, we have extended the derived analytic formula of MDC (of the studied class) to general convection-diffusion problems. The analytic formula is
computed locally within each element according to the mesh size and the values of the associated coefficients in each direction. The numerical results for two-dimensional,
variable coefficients, convection-dominated problems show that although the discrete solution does not coincide with the exact one, it provides stable and accurate solution even on coarse grids. As a result, multigrid-based solvers benefit from these accurate
coarse grid solutions and retained its efficiency when applied for convection-
diffusion equations. Many numerical results are presented to investigate the
convergence of classical algebraic and geometric multigrid solvers as well as Krylov-subspace methods preconditioned by multigrid.
Also, in this thesis, we were concerned with the channel flow, which is an
interesting problem in fluid dynamics. This type of flow is found in many real-life applications such as irrigation systems, pharmacological and chemical operations, oil-