Finite element method with quasi-linearization for solving Bratu's problem

Abstract

This work presented here is the solution of one-dimensional Bratu’s problem. The major aim of this research is to master the techniques used to solve the Bratu problem. The nonlinear Bratu’s problem is first linearised using the quasi-linearization method and then solved by the finite element method using 1. piecewise linear Lagrange polynomials as basis functions 2. piecewise quadratic Lagrange polynomials as basis functions and 3. hierarchical basis functions. Unlike other basis functions like the trigonometric functions, the three basis functions used in this research have an advantage that they have small local support, that is, they are only non- zero on a small portion of the given domain. A comparison of the exact solution and the finite element solutions using Matlab plots and tabulated results is made. The finite element solutions are validated using both Matlab’s bvp4c and the Chebyshev spectral collocation method.