On the convergence of a finite element method

Abstract

Finite element methods (FEM) have become popular enough to have textbooks written about them for science and engineering seniors and/or beginning graduate students. It has become a standard academic offering. Some even believe that the conventional approach to the study of fields influenced by the development of FEM sbould be reconsidered.
The FEM is a computational method used in the approximate solution of ordinary and partial differential equations. Most involved problems of applied physics have used FEMs to approximate solutions to problems in fluid dynamics, structural mechanics, nonlinear optics, and molecular interactions among others.
The main advantage of the FEM over finite-differencing (FD) scheme in the solution of differential equations is the relative ease with which one could deal with boundaries of irregular shapes. Also, one is not limitted to equal intervals in the formation of a grid in the discretization of a problem. When applied to spatio-temporal problems, FEM has the advantage of having the convergence/stability of the time and space variables independent of each other. Unlike finite-differencing schemes for the solution of partial differential equations of the parabolic or hyperbolic type, where the stability of the method depends on the relative graininess of both space and time variables.
Just like finite-difference methods where the number of intervals is increased, FEMs converge faster towards the true solution as the number of elements is increased. Unlike finite-differencing, the FEM involves a number of procedures with at least two of them open to many types of approaches, which differentiates the types of FEMs.
In this investigation, a FEM was applied to a two point boundary value problem and its convergence with the number of elements used was estimated by a mean square error. For comparison the same is done using FD, varying the number of subintervals instead of the number of elements. It is found that FD converges a lot faster than the FEM implemented here which uses only linear interpolating functions. Higher order polynomials are needed to have a comparable accuracy to FD.