# Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations
## 1 Introduction
Various recovery- and residual-based error estimators have been developed for utilization in adaptive-mesh-improvement strategies and for enhancing the accuracy and reliability of finite element method results [20]. [PERSON] and [PERSON] [21] proposed explicit residuals and an improved Zienkiewicz-Zhu (ZZ) error-estimator-based hybrid a posteriori error estimator, applicable in tandem with the finite element technique. [PERSON] et al. [22] presented reliable and efficient residual-based a priori and a posteriori error estimators for mixed finite element methods, and the good performance of the proposed method was confirmed by solving the linear elasticity problem. [PERSON] et al. [23] proposed a mesh-free MLS approach with greater computational efficiency for the static and quasi-static analysis of thin-shell fractures. They utilized error estimations based on gradients to add linear reproducing points in locations needing refinement. A hp-adaptive computational analysis applicable to a variety of solid mechanics problems was carried out by [PERSON] et al. [24]. [PERSON] and [PERSON] [25] presented a MLS interpolation approach with high computational stability and accuracy, in which shape functions at nodes perform the interpolation activity. The effectiveness of the discrete stationary moving least-squares fitting approach, the continuous stationary moving least-squares approach, and the moving least-squares approach, suggested for the element-free Galerkin method, were evaluated by [PERSON] and [PERSON] [26]. They found that that the performance of the discrete stationary moving least-squares approach was better than the other recovery approaches. The mixed-formulations-based finite element approach in conjunction with error estimators is utilized for the adaptive finite element simulation of linear incompressible elasticity problems [27; 28]. Some a posteriori error estimators for nearly incompressible linear elasticity were developed by [PERSON] and [PERSON] [29] using goal-oriented estimations based on the dual weighted residual approach. The application of the MLS-based mesh-less recovery technique for gradient-error recovery in elastic analysis was proposed by [PERSON] et al. [30]. [PERSON] al. [31] performed a probabilistic error analysis on a Gaussian-process-based statistical finite element method (statFEM). The adaptive simulation of magnetized plasma transport in fusion reactors was carried out by [PERSON] et al. [32]. They employed Zienkiewicz-Zhu error estimation and quadrilateral elements in their study. The accuracy and efficiency investigations of finite element formulation and the non-polynomial trigonometric higher-order shear deformation theory (HSDT) used in the vibration analysis of bi-directional functionally graded plates with geometric imperfection and porosity was carried out by [PERSON] et al. [33]. The accuracy-and-robustness study of finite element formulation and the improved first-order shear-deformation theory used in the bending/buckling analysis of bi-directional functionally graded plates with porosity was performed by [PERSON] et al. [34]. [PERSON] et al. [35] proposed a mesh-free radial point interpolation technique to recuperate the field-variable error in the finite element analysis of incompressible elastic problems and showed that error estimators can be successfully implemented for reliable adaptive discretization. According to the review of the relevant literature, it is clearly evident that mesh-free recovery-based error estimation is a relatively new interest in finite element analysis, and few studies have investigated the different mesh-free procedures in mesh-free recovery-based error estimation specifically for incompressible elasticity and large domain changes. Therefore, there is a need to develop mesh-free recovery techniques using different mesh-free procedures, especially for problems of incompressible elasticity and large domain changes.