Elliptic Boundary Value Problems (EBVP) are typically associated with steady-state behavior. It
is a class of problems which do not involve time variable and only depend on space variables instead. One of the famous example of elliptic equation, Poisson's equation which is an elliptic linear inhomogeneous partial differential equation of the second order. It is very useful in resolving problems for both science and engineering applications. Poisson's equation is discretized by using central finite difference scheme. This scheme generates numerical solution which leads to the discrepancy between numerical solution and analytical exact solution. As a consequence, the sources of error arise in the form of numerical error, discretization error, convergence error, and round-off error. The existence of these sources affecting to the numerical parameters of numerical scheme. This problem can be tackled by presenting convergence acceleration techniques. The examples of convergence acceleration techniques conducted on this thesis are Optimum SOR method and Multigrid method. Furthermore, the results of these acceleration techniques are compared with convergence non-acceleration techniques such as Jacobi method and Gauss-Seidel method.