Superconvergence effect in finite element methods for two-point boundary value problems

Approximate methods for solving two-point boundary value problems are considered. The aim of the paper is to explain superconvergence effect in the methods using finite element spaces. The existence of a class of the methods with the superconvergence property is demonstrated. Detailed proofs of superconvergence are presented for the case of the Galerkin method (due to Douglas and Dupont results) and for some example of external method.