# Improvement of the Zienkiewicz-Zhu Error Recovery Technique Using a Patch Configuration
## 1 Introduction
A gradient recovery method, the so-called super-convergent cluster recovery method, was presented by [PERSON] and [PERSON] [18] in which a linear polynomial approximation was obtained by a least-squares fitting to the finite element solution at certain sample points. An error recovery procedure based on the least-squares fitting of the displacement field over an element neighborhood patch configuration was proposed by [PERSON] et al. [19]. [PERSON] et al. [20] proposed an error estimation procedure to accurately estimate the relative eigenvalue errors and to recover the solution error in an automated multilevel substructure method. [PERSON] and [PERSON] [21] presented a super convergence in the L\({}_{2}\) norm using the patch recovery method to recover linear finite element solutions. [PERSON] et al. [22] proposed a hybrid error recovery technique, consisting of the explicit residual and an enhanced Zienkiewicz-Zhu (ZZ) error recovery technique, for the conforming finite element method, and verified that the developed technique was accurate for all meshes. [PERSON] et al. [23] proposed a stress recovery procedure for low-order finite elements in three dimensions. [PERSON] et al. [24] proposed an error estimation based on the element energy projection technique for adaptive finite element analysis. There has also been interest in developing support domain or mesh independent patch recovery techniques. [PERSON] et al. [25] developed element-free Galerkin (EFG) approach-based error recovery techniques for finite element analysis employing mesh independent node patches. From the literature review, it is clear that the quality of recovery-based error estimation depends on the approach for recovering the stress or displacement field that satisfies the equilibrium and boundary conditions. The ZZ recovery technique uses the least-squares fitting of stress field by the same order polynomial as that present in the basis function over a patch of neighborhood nodes. The number of polynomial terms in the basis function should be limited, in order that the number of unknown parameters in the interpolation equation does not exceed the number of independent equations for a particular patch configuration. The interpolation equation should be used with utmost care at boundaries where too few linear independent equations can be established within the patch. The improved ZZ method of error recovery, especially at a boundary, is proposed using a patch configuration (element neighborhood patch) larger than the standard patch (node neighborhood patch). The advantage of the proposed patch configuration is that it provides more sampling points near the boundary and improves the error recovery of boundary regions. The effectiveness and reliability of the improved ZZ recovery approach is assessed through plane elastic and plastic plate problems. The problem domains are discretized with triangular and quadrilateral (regular/irregular) elements with different sizes of regular and irregular subdivisions. We also carried out the least-squares fitting of the displacement field by a higher order polynomial to extract stress over a similar element neighborhood patch configuration. The quality of error estimation using the ZZ recovery of the derivative field was compared with that of error estimation using extracted stress from the recovery of a more accurate displacement field in terms of effectivity, rate of convergence, and adaptively refined meshes.