Global Superconvergence Of Finite Element Methods For Two-point Boundary Value Problem

The convergence order of the nodal values of the finite element solution exceeds the possible global convergence order. We can use the finite element postprocessing technique for the finite element solution in order to obtain a higher convergence order than usual one. In this paper, we obtain the high global accuracy for two point boundary value problem , by using the finite element postprocessing technique on some superconvergence estimates.We have known that the finite element solution can get the convergence accuracy order of m+1 in global interval and of 2m in the nodes. This paper proposes a new approach of useing the postprocessing technique to get the convergence accuracy order of 2m in global interval for m degree element on the basis of the original numerical result. First we give the decomposition of global interval, and use the finite element method to get the values of nodes. Then we use them as boundary values, and keep on solving the equations with finite element method in every subintervals. Finally, we allocate the degree of optimal grid splitting and the degree of optimal finite element in every subintervals in parallel, and obtain the optimal convergence order. The numerical results show that this method has smaller computational amount and higher global accuracy than the refinement method on the same level of grid.This paper is divided into four parts. Section 1 introduces the historical background about postprocessing technique for the finite element. Section 2 provides some preparatory knowledge. In Section 3, we discuss how to apply postprocessing technique exactly in order to obtain the high global accuracy. In the fourth Section, we give the numerical results. In the end of the paper, we draw a conclusion of this work and make some comments on the prospect of the moving point method.