A Comparative Study of finite Difference and Galerkin Finite Element Methods for Solving Boundary Value Problems (European Journal of Pure and Applied Mathematics, 2025)
Many applications in engineering cannot be solved analytically, major difficulty in the study of partial differential equations is that it is often impossible to obtain analytical solutions. Therefore, various numerical methods for solving partial differential equations have been proposed by related researchers, such as the finite difference method (FDM), finite element method (FEM), finite volume method (FVM), etc. The earliest is the FDM, which approximates the differential equations by using a local Taylor expansion. The finite element method (FEM) is a numerical technique for solving problems which are described by partial differential equations or can be formulated as functional minimization. A domain of interest is represented as an assembly of finite elements.This paper presents a qualitative comparative study of FDM and Galerkin finite element method (GFEM) to show the advantages and disadvantages of these methods in solving boundary value problems. Several numerical experiments conducted for comparisons purpose. The results reveal that the GFEM is an alternative method for solving several types of boundary value problems.
Keywords: Boundary value problem; Finite difference method; Galerkin Finite element method; Poisson's equation