# Improvement of the Zienkiewicz-Zhu Error Recovery Technique Using a Patch Configuration
## 1 Introduction  
The finite element method is the most used numerical tool for solving industrial problems. Recently, there has been interest in overcoming the drawbacks of the finite element analysis method, and therefore improving the quality of finite element analysis results using a priori and a posteriori error estimators. A comprehensive review on the diverse applicability of the finite element approach is given by [PERSON] et al. [1]. The different types of error estimation techniques have been reviewed by [PERSON] and [PERSON] [2]. [PERSON] et al. [3] showed that the Zienkiewicz-Zhu error estimation was effective, convergent, and asymptotically exact when the exact stress boundary conditions were implemented. The classical super-convergent patch recovery (SPR) technique or ZZ recovery technique [4] has been proposed to recover the lost accuracy and continuity of the stress field by interpolating from a stress surface fitted to the super-convergent stress points in a node neighborhood patch. They emphasized that the error estimation would be asymptotically exact if the recovery of the finite element solution was super convergent. An in-depth discussion onimproving the SPR technique was presented by [PERSON] et al. [5]. [PERSON] and [PERSON] [6] proposed a recovery procedure (SPR-E technique) for improving the accuracy of the recovered stresses by imposing additional constraints on the equilibrium equations. [PERSON] and [PERSON] [7] extended the ZZ recovery technique (SPR-EB technique) by including the squares of the residual of equilibrium equations and natural boundary conditions. A new conjoint polynomial for interpolation of the local patch stresses on the element was proposed which improved the derivative field within the element as compared with standard nodal interpolation. [PERSON] and [PERSON] [8] presented a post-processing technique for determining more accurate solutions by fitting a higher order polynomial expansion to the computed solution at super-convergent points in the element patch that represented the union of the considered element and its neighbors. [PERSON] et al. [9] proposed a patch recovery scheme for all stress components and coupling of the stress components was achieved through the equilibrium equations. [PERSON] et al. [10] proposed a super-convergent stress recovery technique (LP technique) for obtaining stress interpolation polynomials over a patch of elements formed around a vertex node using the virtual work form of the equilibrium residual, while, employing a similar idea, [PERSON] et al. [11] proposed a recovery technique (SP technique) assuming stress field over a single element. The Trefftz patch recovery (TPR) technique using Trefftz fields was proposed by [PERSON] [12] to obtain patch interpolation polynomials which satisfied the internal equilibrium and compatibility equations. [PERSON] et al. [13] presented a modified SPR technique, the so-called SPR-C technique using the appropriate constraint equations, to produce stress interpolation polynomials that exactly satisfied the internal and boundary equilibrium equations. The explicit-type recovery error estimator in energy norm was proposed by [PERSON] et al. [14] for the linear elasticity problem using a smooth solution. [PERSON] et al. [15] developed a recovery-based error estimation technique for the [PERSON] gradient elasticity theory to avoid the occurrence of singularities in the stress and strain fields. The super-convergent patch recovery (SPR) method was modified by [PERSON] et al. [16] to improve the method accuracy and stability using an integration point as a sampling point, additional nodes, and weighted average procedure. [PERSON] and [PERSON] [17] developed the polynomial preserving recovery (PPR) method to recover the accuracy in a finite element solution.