Comparison Of Finite Element Method And Finite Difference Method For Boundary Value Problems

Abstract

The main purpose of this thesis is to analysis Rayleigh-Ritz method, Galerkin method and nite di erence method for solving two-point boundary value problems of ordinary di erential equa- tions. The two point boundary value problems have great importance in chemical engineering, de ection of beams and other elds. Rayleigh-Ritz and Galerkin nite element methods are applied for solving inhomogeneous second-order ordinary di erential equations. Rayleigh-Ritz method needs varitional formulation and minimization . But Galerkin's method needs weak for- mulation which leads to solving linear algebraic systems. Both methods are based on piece-wise interpolation polynomials on domain and trial solutions. The methods are used to calculate the unknown parameters of the trial solutions and then form a trial function which is used to calcu- late the numerical solution. To obtain a good approximation, the choice of the base function is important and to improve the approximation the number of base functions should be increased. In addition,we study the convergence, stability, and consistency of each method. Finite di erence method is based on domain discretization. In nite di erence method, we rst discretize the domain of the problem into equal mesh points and substitute the second order central di erence. Using the discretized mesh points we construct systems of linear equations and form tri-diagonal matrix, then using Thomas algorithms we calculate the solutions and also derive the errors at each mesh points. The error analysis of FDM is based on Taylor's theory. For nite di erence method we take di erent mesh lengths x. The convergence, stability and consistency depend on the mesh length. As the mesh length tends to zero, the convergence, stability and consistency is better. For Rayleigh-Ritz method and Galerkins method, the conver- gence, stability and consistency depends on the trial solutions. As the degree of interpolating polynomial increases the maximum absolute error tends to zero, which means the FEM is more convergent, stable and consistent. Rayleigh-Ritz method and Galerkins methods are e cient and faster. Their stability, consistency and convergence is better than that of nite di erence method.