Superconvergence and recovery type a posteriori error estimation for hybrid stress finite element method

TL;DR

This paper investigates superconvergence and recovery-based a posteriori error estimators for a hybrid stress finite element method in linear elasticity, demonstrating their effectiveness through theoretical analysis and numerical validation.

Contribution

It establishes uniform superconvergence rates and asymptotic exactness of error estimators for the 4-node hybrid stress quadrilateral element, extending understanding of error behavior in finite element analysis.

Findings

Superconvergence order of O(h^{1+min{α,1}}) for gradients and stresses.

A posteriori estimators are asymptotically exact.

Numerical results confirm theoretical predictions.

Abstract

Superconvergence and a posteriori error estimators of recovery type are analyzed for the 4-node hybrid stress quadrilateral finite element method proposed by Pian and Sumihara (Int. J. Numer. Meth. Engrg., 1984, 20: 1685-1695) for linear elasticity problems. Uniform superconvergence of order $O(h^{1+\min\{\alpha,1\}})$ with respect to the Lam\'{e} constant $\lambda$ is established for both the recovered gradients of the displacement vector and the stress tensor under a mesh assumption, where $\alpha>0$ is a parameter characterizing the distortion of meshes from parallelograms to quadrilaterals. A posteriori error estimators based on the recovered quantities are shown to be asymptotically exact. Numerical experiments confirm the theoretical results.