New convergence analysis for assumed stress hybrid quadrilateral finite element method

Binjie Li; Xiaoping Xie; Shiquan Zhang

doi:10.3934/dcdsb.2017153

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2017, Volume 22, Issue 7: 2831-2856. Doi: 10.3934/dcdsb.2017153

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New convergence analysis for assumed stress hybrid quadrilateral finite element method

School of Mathematics, Sichuan University, Chengdu 610064, China

* Corresponding author: Xiaoping Xie, xpxie@scu.edu.cn
* Corresponding author: Xiaoping Xie, xpxie@scu.edu.cn

Received: May 31, 2016

Revised: February 28, 2017

Published: October 2017

This work was supported by Major Research Plan of National Natural Science Foundation of China (91430105) and National Natural Science Foundation of China (11171239,11401407).

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Abstract

New error estimates are established for Pian and Sumihara's (PS) 4-node assumed stress hybrid quadrilateral element [T.H.H. Pian, K. Sumihara, Rational approach for assumed stress finite elements, Int. J. Numer. Methods Engrg., 20 (1984), 1685-1695], which is widely used in engineering computation. Based on an equivalent displacement-based formulation to the PS element, we show that the numerical strain and a postprocessed numerical stress are uniformly convergent with respect to the Lamé constant $λ$ on the meshes produced through the uniform bisection procedure. Within this analysis framework, we also show that both the numerical strain and stress are uniformly convergent on meshes which are stable for the $Q_1-P_0$ Stokes element.

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Mathematics Subject Classification: 65N12, 65N15, 65N30.

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Figure 1. A stable mesh for $Q_1\!-\!P_0$ Stokes element

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Figure 2. $\widehat\chi\circ F_K^{-1}$ for some $K\in\mathcal T_{2h}$

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