Comparison of finite element method and the cubic spline to solve Boundary Value problems

Abstract

Many physical problems are modeled by Boundary value problems and the solution sometimes may not be easily obtained analytically. In that case, we use different numerical methods. Among these numerical methods, in this paper, we present the Finite difference method, the Galerkins finite element method, and the cubic spline method for solving boundary value problems. They have great importance in Science and Engineering. Finite difference method is based on domain discretization. We first discretize the domain in to equal sub-intervals and then substitute the differentials with finite difference schemes. The tri-diagonal matrix system formed is solved using Thomas algorithm or any other Numerical Method. The Galerkin’s finite element method for solving Boundary value differential equation is also examined. The Galerkin’s finite element method needs base functions using Lagrange polynomials and it needs to define residue with weak formulation in the boundary conditions. The tri-diagonal matrix formed is carried out using Thomas algorithm or LU decomposition. We also choose cubic spline functions to develop the numerical solution of boundary value problems and use linear combination of cubic spline base functions to approximate the solution. The resulting linear systems of equations is solved using a tri-diagonal solver. To obtain a good approximation, the choice of the base function is important and to improve the approximation the number of base function is increased,. Convergence of the method shown through standard convergence analysis, finally, we compare the Galerkin’s finite element method and cubic spline method with finite difference method, and then the advantage of the former is illustrated. For this, we select a second order boundary value problem and then discritize the domain of interest in to different nodal points and apply the three methods to determine the solutions. As the degree of the Lagrange polynomial increases, the maximum absolute error tends to zero, which means the Galerkins finite element method is more convergent, stable and consistent. The absolute errors for test examples are estimated, the comparison of approximate values, exact values and absolute error at the nodal points are shown using tables and graph, Further shown that the Galerkin’s finite element method and the cubic spline methods give more accurate solution comparing with finite difference method.