## 5 Benchmark Example Applications
### Application to Compressible Elastic Plate
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multicolumn{10}{c}{**Linear Triangle (Regular)**} & \multicolumn{10}{c}{**Quadratic Triangle (Regular)**} \\ \hline \multicolumn{10}{c}  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Elements**} & \multirow{3}{*}{**DOF**} & \multicolumn{9}{c}{**Quadratic Triangle (Irregular)**} \\ \cline{3-14}  & & \multicolumn{9}{c}{**Displacement Recovery**} & \multicolumn{9}{c}{**Stress Recovery**} \\ \cline{3-14}  & & \multicolumn{9}{c}{**FEM Error (Disp. )**} & \multicolumn{1}{c}{**\(\theta\)**} & \multicolumn{1}{c}{**FEM Error (Train)**} & \multicolumn{1}{c}{**\(\theta\)**} & \multicolumn{1}{c}{**FEM Error (Train)**} & \multicolumn{1}{c}{**\(\theta\)**} & \multicolumn{1}{c}{**FEM Error (Stress)**} & \multicolumn{1}{c}{**\(\theta\)**} & \multicolumn{1}{c}{**FEM Error (Stress)**} & \multicolumn{1}{c}{**\(\theta\)**} \\  & & \multicolumn{9}{c}{**(\(\times\)10\({}^{-3}\))**} & \multicolumn{The adaptive analysis results are obtained for mesh required for target error limit for stresses, displacement, and strains. Tables 6 and 7 show global errors (FEM and processed), the number of element (N), and degrees of freedom (DOF) in refined meshes, obtained from the linear and quadratic elements of the initial mesh, to bring accuracy to the target level. The required adaptive meshes for aimed accuracy in various L\({}_{2}\)-norms, namely stresses, displacement, and strains, using the least square stress, and displacement recovery-based error estimation for linear and quadratic elements are portrayed in Figures 4 and 5. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Recovery Type/Initial**} & \multicolumn{4}{c}{**Displacement Error Recovery**} & \multicolumn{4}{c}{**Stress Error Recovery**} \\ \cline{2-10}  & **FEM** & **Error (\%)** & \multicolumn{2}{c}{**Adaptive Mesh**} & \multicolumn{2}{c}{**FEM**} & \multirow{2}{*}{**Error (\%)**} & \multicolumn{2}{c}{**Adaptive Mesh**} \\ \cline{3-3} \cline{5-10}  & **Error (\%)** & & **N** & **DOF** & & **Error (\%)** & & **N** & **DOF** \\ \hline L\({}_{2}\)-Norm-Displacement & & 15.88 & 16.13 & 239 & 282 & - & - & - & - \\ (N = 28, DOF = 46) & & 15.88 & 16.13 & 239 & 282 & - & - & - & - \\ L\({}_{2}\)-Norm-Strain & & 7.78 & 7.74 & 2520 & 2678 & - & - & - & - \\ (N = 580, DOF = 652) & & 7.45 & 7.43 & 2154 & 2304 & 7.45 & 9.75 & 3090 & 3296 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Global errors (FEM and processed, L\({}_{2}\)-Norm) and number of element (N) with DOF of adaptively refined meshes (considering displacement and stress recovery-based analysis (linear triangle, 3% target error). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Recovery Type/Initial**}} & \multicolumn{4}{c}{**Displacement Error Recovery**} & \multicolumn{4}{c}{**Stress Error Recovery**} \\ \cline{2-10}  & **FEM** & **Error (\%)** & \multicolumn{2}{c}{**Adaptive Mesh**} & \multicolumn{2}{c}{**FEM**} & \multirow{2}{*}{**Error (\%)**} & \multicolumn{2}{c}{**Adaptive Mesh**} \\ \cline{3-3} \cline{5-10}  & **Error (\%)** & & **N** & **DOF** & & **Error (\%)** & & **N** & **DOF** \\ \hline L\({}_{2}\)-Norm-Displacement & & 0.71 & 2.25 & 94 & 430 & - & - & - & - \\ L\({}_{2}\)-Norm-Strain & & 4.34 & 8.24 & 750 & 3162 & - & - & - & - \\ L\({}_{2}\)-Norm-Stress & & 4.12 & 8.28 & 477 & 2036 & 4.12 & 3.91 & 203 & 904 \\ (N = 28, DOF = 146) & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Global errors (FEM and processed, L\({}_{2}\)-Norm) and number of element (N) with DOF of adaptively refined meshes considering displacement and stress recovery based analysis (quadratic triangle, 1% target error).