Similarity, For Boundary Value Problems And The Initial Value Problem The Finite Element Superconvergence

In this paper,based on an improved orthogonal expansion in an clement , using the new idea of Ref.[3] ,a new error expression of n-degreeHermite finite element approximation to one-dimensional 4-degrec2-point bounded problem and 2-degree ordinary differential problem, and then optimal order superconvergence for their first derivatives is obtained. Moreover, we get the same result of their optimal order superconvergence.The paper organizes as follows:In section l:Use the new idea of Ref.[3], we discuss one-dimensional 4-dcgree 2-point bounded problem ,the new error result and firstderivatives of n-degree Hermite finite element approximation is obtained, i.e.In addition, the computing result is the same as the theory.In section II:We discuss 2-degree ordinary differential problem ,the new error result and First derivatives of m-degree Hermite finite elementapproximation is obtained, i.e.In addition, the computing result is the same as the theory.All in all, we obtain the same superconvergence result of one-dimensional 4-degree 2-point bounded problem and 2-degree ordinary differential problem.