A computational study of finite element methods for second order linear two-point boundary value problems

A computational study of five finite element methods for the solution of a single second order linear ordinary differential equation subject to general linear, separated boundary conditions is described. In each method, the approximate solution is a piecewise polynomial expressed in terms of a B-spline basis, and is determined by solving a system of linear algebraic equations with an almost block diagonal structure. The aim of the investigation is twofold: to determine if the theoretical orders of convergence of the methods are realized in practice, and to compare the methods on the basis of cost for a given accuracy. In this study three parametrized families of test problems, containing problems of varying degrees of difficulty, are used. The conclusions drawn are rather straightforward. Collocation is the cheapest method for a given accuracy, and the easiest to implement. Also, for solving the linear algebraic equations, the use of a special purpose solver which takes advantage of the structure of the equations is advisable. 1. Introduction. Finite element methods for two-point boundary value problems for a single ordinary differential equation may take many forms. The simplest type of method is probably collocation (3) where an approximate solution is sought in some finite space subject to the constraint that it satisfy the differential equation at certain specified points. This type of method has been applied very successfully in the case of mixed order systems of boundary value problems; see, for example, (1). Other methods can be described based on the standard L2-Galerkin approximation (9), or on a combination of this and the collocation approach (7), (13), (25). In addition, different weak formulations of the boundary value problem lead to two other techniques, the Hx- and //"'-Galerkin methods (12), (14), (15), (17). How one would apply these methods to a system of differential equations, with the exception of the collocation case, is not immediately clear. In addition, it is not apparent what the relative advantages are of these various methods, even when applied to a single equation. In this paper we restrict our attention to the case of a single linear differential equation. Our aim is to investigate numerically five finite element techniques for a second order linear two-point boundary value problem, with separated boundary conditions of a general nature. In particular, we consider (i) the organization of the methods; (ii) the treatment of the boundary conditions;