Abstract
Elliptic Partial Differential Equations of second order have been studied using some numerical methods. This type of differential equations has specific applications in physical and engineering models. In most applications, first- order and second-order formulas are used for the derivatives. In this work higher order formulas such as: seven-points and nine-points formulas are used. Using these formulas will transform the partial differential equation into finite difference equations. To solve the resulting finite difference equations the following iterative methods have been used: Jacobi method, Gauss-Seidel method, Successive Over- Relaxation method (SOR) and Multigrid method.
In this thesis, we found that multigrid methods are the most efficient among all other methods. The execution time for multigrid methods is of order three while the other methods is of order five.

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Numerical Methods for Solving Elliptic