Numerical Computations for Some Boundary Value Problems using Finite Element Method

Abstract

The finite element method (FEM) is a computational technique for obtaining approximate solutions to the differential equations that arise in scientific and engineering applications. Rather than approximating the differential equation directly, the finite element method utilizes a variantional problem that involves an integral of the differential equation over the domain problem. This domain is divided into a number of subdomains called finite elements and the solution of the differential equation is approximated by a simpler polynomial function on each element . In this thesis, we use the finite element method to solve some boundary value problems. We approximate the solution using the hat function as a polynomial of degree one . We approximate the resulting integrals in two different ways: one way we use the Trapezoidal rule, and the other way we use Simpson's rule we study a two – point boundary value problem with regular and singular coefficients also, we use the finite difference method (FDM) for solving these problems for comparison reasons. We use MATLAB to show the comparative results between the exact and numerical solutions.