Differential equations, and especially the partial deferential equations (PDEs), constitute a very important part of the majority every day problems. One of the most popular computational method for solving partial differential equations is the Finite Element Method (FEM), and it is based on the discretization of space via mesh. Meshing, plays an important role in the overall performance of the method and the accuracy of the solution. In this thesis, finite element method was applied , using different meshes for different refinement levels with various order in polynomial approaches, in order to solve the stationary Poisson Equation. For the simulation, different geometries were used, starting from a simple unit cube to more complex ones. Furthermore, for the unit cube, domain partition was also implemented using structured grids. The overall work was done via a modular toolbox called DUNE( Distributed and Unified Numerics Environment).