Superconvergence patch recovery for the gradient of the tensor-product linear triangular prism element

Jinghong Liu,Yinsuo Jia

doi:10.1186/1687-2770-2014-1

Abstract

In this article, we study superconvergence of the finite element approximation to the solution of a general second-order elliptic boundary value problem in three dimensions over a fully uniform mesh of piecewise tensor-product linear triangular prism elements. First, we give the superclose property of the gradient between the finite element solution and the interpolant Πu. Second, we introduce a superconvergence recovery scheme for the gradient of the finite element solution. Finally, superconvergence of the recovered gradient is derived.

Highlights

Superconvergence of the gradient for the finite element approximation is a phenomenon whereby the convergent order of the derivatives of the finite element solutions exceeds the optimal global rate
We studied the superconvergence patch recovery (SPR) technique introduced by Zienkiewicz and Zhu [ – ] for the linear tetrahedral element and proved pointwise superconvergent property of the recovered gradient by SPR
We introduce a tensor-product linear polynomial space denoted by P, that is, q(x, y, z) =

Summary

Introduction

Superconvergence of the gradient for the finite element approximation is a phenomenon whereby the convergent order of the derivatives of the finite element solutions exceeds the optimal global rate. We studied the superconvergence patch recovery (SPR) technique introduced by Zienkiewicz and Zhu [ – ] for the linear tetrahedral element and proved pointwise superconvergent property of the recovered gradient by SPR. For the linear tetrahedral element, Chen and Wang [ ] discussed superconvergent properties of the gradients by SPR and obtained superconvergence results of the recovered gradients in the average sense of the L -norm. Chen [ ] and Goodsell [ ] derived superconvergence estimates of the recovered gradient by the L -projection technique and the average technique, respectively.

Results

Conclusion