A Thesis On Analysis Of Finite Element Method And Finite Difference Method For Solving Poisson's Equation In 2d

Abstract

In this thesis, we focus on analysis of Finite Element Method and Finite Di erence Method for the numerical solution of Poisson's equation in 2D with Dirchlet boundary conditions. The fundamental idea of the nite element method is the replacement of continuous func- tions by piecewise approximations, usually polynomials. The nite element method has now reached a very sophisticated level of development, so much so that it is applied rou- tinely in a wide variety of application areas, a general description of the steps involved in obtaining a solution to a Poisson's equation is provided. This description includes discussion of the types of elements available for a nite element method solution such as triangulations. Weak formulation reduce the continuity requirements on the approxi- mation (or basis functions) functions there by allowing the use of easy to construct and implement polynomials. Weak forms give relatively very accurate results with the mesh re nement, which are extremely good for engineering simulations. Improving the accu- racy of a solution in weak formulations depend upon the type of problem we are solving. In poisson's equation only mesh re nement is good enough to get accurate results. The accuracy can also be improved by using higher-order shape functions. Finite di erence methods are numerical methods for solving di erential equations in which nite di erence approximate the derivatives. Discretization, convergence, consistence, and stability anal- ysis of the method for solving poisson's equation in 2D is discussed. Numerical examples are provided using basic Matlab codes. The study shows that the error decrease as the discretization of the element increases on its domain and convergence ensured by suitable mesh re nement and nite element method is better in terms of convergence than nite di erence method for solving poisson's equation in 2D.