Finite element solutions to boundary value problems

Abstract

The finite element solution of certain two-point boundary value problems is discussed. In order to obtain more accuracy than the linear finite element method can give, an order-h[superscript 4] global superconvergence technique is studied. This technique, which uses a quasi-inverse of the Rayleigh-Ritz-Galerkin (finite element) method, is motivated by the papers of C. de Boor and G. J. Fix [14] and P. 0. Frederickson [25]. The Peano kernel theorem is generalized and used to approximate the rate of convergence of the global superconvergence. Following Sard’s theory on best quadrature formulae [50], with some generalization, several quadrature formulae are derived. These quadrature formulae are shown to be consistent, and have some advantages over those obtained by Herbold, Schultz and Varga [34]. For solution of large linear systems which result from the finite element method, LU decomposition (Gaussian Elimination Method) is fast and accurate. However, when it comes to a singular or a nearly singular system, LU decomposition fails. The algorithm FAPIN developed by P. 0. Frederickson for 2-dimensional systems is able to solve singular systems as we demonstrate. We found FAPIN will work more efficiently in 1-dimensional case if we replace the DB[subscript q] approximate inverse C, developed by Benson [3], with other approximate inverses. For the sake of verifying the theory, appropriate numerical experiments are carried out.