|dc.description.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  and P. 0. Frederickson . 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 , 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 .
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 ,
with other approximate inverses.
For the sake of verifying the theory, appropriate numerical
experiments are carried out.||