A Statistical Approach for Error Estimation in Adaptive Finite Element Method

Abstract

In the adaptive finite element method, the purpose is to create an optimal mesh with minimum degrees of freedom while producing an error value that is lower than the aim error. For this purpose, the estimation of discretization error at different points of the domain is required. In the Zienkiewicz and Zhu error estimator, the stress values of Gauss points is improved using superconvergent patch recovery and this improved solution is compared with the FEM solution to estimate the error. The regions with higher gradients of stress field produce larger values of estimated error. But the optimal mesh usually is attained in several steps of remeshing which leads to a high computational cost. In the present study, to track the regions with high gradients of stress, the statistical distribution of the stress values at Gauss points around a node is compared with the uniform distribution function, and the difference between these two distributions is taken as the error estimator. In this technique, at points for which the stress gradient is high, the mesh is automatically refined. The efficiency of this technique is that it finds the optimal mesh quickly in fewer steps. This advantage is demonstrated by several numerical examples.