Gedif, TsegayeDaniel, Gloria2018-07-162023-11-042018-07-162023-11-042016-06http://etd.aau.edu.et/handle/123456789/8687This project is about the Galerkin method and its variants.Its variation are the Petrov- Galerkin method,the generalized Galerkin method and Conjugate gradient method. The Galerkin method provides numerical method to solving elliptic boundary value problem and its application in error estimate of Galerkin solution of linear elliptic boundary value problem. This project also including the numerical solution of a one dimensional poisson with boundary condition Idea of Galerkin discretization Replace V in a(u,v)=`(v) 8u; v 2 V with a _nite dimensional subspace VN, is also viewed and some examples are used to illustrates some numerical methods for solving boundary value problems for the Partial Di_erential Equations(PDEs)to obtained the result. Key works: Dirichlet Boundary condition - Galerkin method, Poisson equation, Numerical solution using MatlabenDirichlet Boundary condition - Galerkin methodPoisson equationNumerical solution using MatlabThesis