## 6 Discussion  
This study presents the a posteriori error estimations of finite element solutions based on recovery of displacement and stress. The benchmark examples of compressible and incompressible elastic bodies are implemented to demonstrate the effectiveness and efficiency of the recovery techniques. The error estimation techniques are based on the least square fitting of higher order polynomials to stress and displacement in a patch comprising of node/elements surrounding and including the node/elements under consideration. The error estimations at global and elemental levels have been presented in the L\({}_{2}\)-norms of quantity of interest, i.e., displacement, strain, and stress. The quality of error estimation in a posteriori recovery techniques are enumerated and matched in terms of convergence properties, effectivity, and adaptively updated meshes for predefined precision. The computational results of error convergence and effectivity with various densities of triangular meshing are given in Table 1, Table 2, Table 3, Table 4, Table 5, Table 8, and Table 10 respectively for elastic square plate, elastic plate with circular inclusion, and incompressible elastic square plate. It is clear from the tables that the order of error obtained with displacement and stress recovery techniques is much lower than the FEM solution error, and the error convergence is also higher when compared to the FEM solution error with an increase in \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Quadratic Triangle (Regular)**} & \multicolumn{4}{c}{**Quadratic Triangle (Irregular)**} \\ \cline{2-10}
**Mesh** & \multicolumn{3}{c}{**Displ. Recovery**} & \multicolumn{3}{c}{**Stress Recovery**} & \multicolumn{3}{c}{**Elements**} & \multicolumn{3}{c}{**DOF**} & \multicolumn{3}{c}{**Displ. Recovery**} & \multicolumn{3}{c}{**Stress Recovery**} \\ \cline{2-10}
**Size** & **FEM** & **Error** & **Error** & **Error** & **Error** & **Error** & **Error** & **Error** & **(Stress)** & \(\theta\) \\  & **Error** & **(\(\times\)10\({}^{-3}\))** & **(\(\times\)10\({}^{-3}\))** & **(\(\times\)10\({}^{-3}\))** & & & **(\(\times\)10\({}^{-3}\))** & & **(\(\times\)10\({}^{-3}\))** & & **(\(\times\)10\({}^{-3}\))** & \\ \hline \(\frac{1}{2}\) & 46.78 & 23.13 & 0.9132 & 24.66 & 0.8735 & 28 & 146 & 39.72 & 27.44 & 0.8196 & 32.23 & 0.4999 \\ \(1/16\) & 16.13 & 3.11 & 0.9693 & 4.82 & 0.9472 & 340 & 1470 & 6.98 & 3.01 & 0.8554 & 3.20 & 0.8517 \\ \(1/32\) & 8.06 & 0.95 & 0.9746 & 1.69 & 0.9667 & 1349 & 5608 & 2.83 & 1.24 & 0.8699 & 1.36 & 0.8461 \\ \hline Conv. & 0.9818 & 1.7805 & \multicolumn{3}{c}{1.4952} & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Error in stress (L\({}_{2}\)-Norm) and global effectivity (\(\theta\)) in incompressible elastic square plate example considering displacement and stress recovery technique for Quadratic Triangular mesh (Regular/irregular). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Displacement Error Recovery**} & \multicolumn{4}{c}{**Stress Error Recovery**} \\ \cline{2-10}
**Recovery Type/Element Type** & **FEM Error** & **Error** & \multicolumn{2}{c}{**Adaptive Mesh**} & \multicolumn{2}{c}{**FEM**} & \multicolumn{2}{c}{**Adaptive Mesh**} \\ \cline{4-10}  & **Error** & **N** & **DOF** & **Error** & **N** & **DOF** \\ \hline L\({}_{2}\)-Norm-Stress (initial quadratic triangular mesh: N = 1349, DOF = 5608, 1\% target error) & 4.11 & 4.63 & 9666 & 39166 & 4.11 & 3.48 & 8392 & 33970 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Global errors (FEM and processed, L\({}_{2}\)-Norm) and number of elements (N) with DOF of adaptively refined meshes considering displacement and stress recovery-based analysis. Figure 9: Adaptive mesh required for 1% target error (L\({}_{2}\)-Norm) incompressible elastic square plate example solution variables considering displacement and stress recovery techniques (Initial regular quadratic triangular mesh: N = 1349, DOF = 5608). the element density.