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.