## 1 Introduction  
The finite element method is the most used computational tool for analyzing industrial problems. However, the errors are inherent in the finite element solution due to discretization of the problem area into subdomains. The other sources of errors in the finite element solution are due to the mathematical model for the problem definition, the algorithm employed for solving system equations, and the method employed for computing derivatives of state variable and round-off due to the machine precision. The attention of error control in the finite element solution has been on the discretization error which is due to incomplete satisfaction of the governing equations and boundary conditions. The recent interest in the finite element method is to minimize the discretization error which is due to incomplete satisfaction of the governing equations and boundary conditions [1]. The review on various applications of the finite element method are presented in [2]. Though advancement are made in the finite element method, there are areas which needed further attention [3]. Several techniques are developed to recover the field variables or gradients and to improve the accuracy of finite element solution. The different types of error estimation techniques are developed for finite element analysis [4]. Among the developed error estimation procedures, the post-processed or recovery type error estimation techniques are easy to implement in the codes and provide local errors as well [5]. In a posteriori error estimation, the recovery of finite element solution error using the least squares interpolation technique is made by fitting a higher order polynomial to the field variable or field variable derivatives over a suitable patch. The difference between the field variable or field variable derivatives from the finite element results and projected field variable or field variable derivatives from the recovery analysis results provides the measure of the error which should be represented in appropriate scale measure or norms. The energy norms and L\({}_{2}\)-norm are commonly used. The energy norms represent the error in the rate of energy dissipation while the L\({}_{2}\)-norm of field variable or field variable derivatives express straightforward physical meaning. The super-convergent patch recovery (SPR) technique or so-called Zienkiewicz and Zhu (ZZ) recovery technique recovers the error in the stress field by interpolating the stress fields over the super-convergent stress (Gauss) points [6]. They have estimated stress recovery-based recovered error in energy norm. In the present study, the stress and displacement-based recovered errors are computed in L\({}_{2}\)-norm. The alternative patch recovery scheme recovers stress fields by incorporating the equilibrium and boundary conditions [7]. [PERSON] and [PERSON] [8] have presented a recovery by equilibrium in patches (REP) recovery technique. The ZZ recovery (SPR) method is modified by [PERSON] et al. [9] by considering the integration point as a sampling point and using additional nodes and weighted average procedure. The improvement of the SPR technique (SPR-C technique) is proposed by [PERSON] et al. [10] using additional constraint equations. The recovery of finite element linear elasticity exact solutions for nonconforming elements using the residual method considering the set of Neumann data is due to [PERSON] and [PERSON] [11]. [PERSON] and [PERSON] [12] have proposed the recovery of the exact solution of the displacement field and strains using the energy predictions over the element patches. [PERSON] et al. [13] have presented an explicit-type recovery technique for the linear elasticity problem using smooth solution. [PERSON] and [PERSON] [14] have used best-t procedure to predict the optimal stress sampling points for the patch recovery of nodal stresses. They have found that Gaussian quadrature points are better candidates for patch recovery in comparison to Barlow points.