Gradient Reconvery Method And Superconvergence Of Crouzeix-Raviart Element For Second-order Elliptic Problem

The Crouzeix-Raviart element(CR element)was originally proposed by Crouzeix and Raviart when they studied the Stokes equation for an incompressible viscous fluids.This paper focuses on the CR finite element method of elliptic equations and investigates its superconvergence and gradient reconstruction method.The first part of the paper investigates the superconvergence of the CR element for the second-order elliptic problems with variable coefficients.On the uniform triangular mesh,in which any two adjacent triangles form a parallelogram.Based on the equivalence between the CR element method and the lowest order RT element method,the convergence order of the new numerical stress approximating the exact stress is proved to be 3/2.In the second part,we proposed and investigated the Superconvergence Cluster Recovery method(SCR)for the CR element.The propesed recovery method reconstructs a C0 linear gradient.A linear polynomial approximation is obtained by a least square fitting to the CR element approximation at certain sample points,and then taken derivatives to obtain the recovered gradient.The SCR recovery operator is superconvergent on uniform mesh of four patterns.It is also verified that the error estimator based on the SCR method is asymptotically accurate by applying to the adaptive finite element method to singular problem and variable coefficient problem.