Formulation of Finite Element Method for 1D and 2D Poisson Equation

The Finite Element Method (FEM) introduced by engineers in late 50's and 60's is a numerical technique for solving problems which are described by Ordinary Differential Equations (ODE) /Partial Differential Equations (PDE) with appropriate boundary/initial conditions or to solve problems that can be formulated as a functional minimization. FEM provides greater flexibility to model complex geometries. It can handle general boundary conditions and variable material properties. It has a solid theoretical foundation which gives added reliability and makes it possible to mathematically analyze and estimate the error in the approximate solution. This paper gives an introduction and methodology to solve a PDE using FEM in 1D and 2D in the simplest way possible such that the young researchers who has less mathematical or engineering background can also understand this technique. Only Poisson equation is solved in this paper. The result of the solution the PDE‟s is also shown computationally using the open source software of FEniCS.