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

Abstract

Abstract. 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.

Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations

“…Method [28] requires a parameter for the construction of refinement intervals. The L 2 norm value of the approximation error is smaller than those reported in [28,65].…”

Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations

“…Method [28] requires a parameter for the construction of refinement intervals. The L 2 norm value of the approximation error is smaller than those reported in [28,65]. with boundary conditions y(−1) = −2, y(1) = 0 and > 0 is a parameter.…”

Almost block diagonal linear systems: sequential and parallel solution techniques, and applications

“…Numerical experiments reported in [113,114] demonstrate abdpack's superiority over the version of solveblok used in colnew [14]. ABD systems where the blocks overlap in rows were discussed in [67,92] in the context of finite element methods, and a modified alternate row and column elimination scheme for such systems is implemented in rowcol [66]. 9/8/2006 10:50 PAGE PROOFS nla97-39 4.1.6.…”