Numerical Solution to Two-point Boundary Value Problems with Neumann Boundary Conditions using Galerkin Finite Element Method

Abstract

In this study, Galerkin finite element method (FEM) has been developed to approximate the solution of both second-order linear with constant and non-constant coefficients, and nonlinear second-order two-point BVP of ordinary differential equations with Neumann boundary conditions. Lagrange linear piece-wise polynomials have been used as basic functions. Linear second order two-point boundary value problem (BVP) of ordinary differential equations (ODEs) with non-constant coefficient was solved by applying Gauss quadrature 3-point rule in the Galerkin FEM. For the nonlinear BVP, the Newton’s method was used with the Galerkin FEM. The errors in approximations have been studied, noting that for this method, errors in the approximations reduce with decreasing element or step size. The convergence and consistency of Galerkin FEM applied to the linear and nonlinear second-order boundary value problems of ordinary differential equations have been discussed. The results have been presented in a number of tables and illustrated using graphs, all generated using MATLAB. Basing on the results from the simulations, it was found that the method was stable, convergent and consistent since further reduction of element or step sizes produced significant reduction in the error of all test problems. Thus, the developed method performs well with linear and nonlinear two point BVPs.