## 5 Benchmark Example Applications
### Application to Incompressible Elastic Plate  
The mixed formulation involving displacement and pressure is used for incompressible elastic analysis [25]. The equilibrium and boundary condition for the incompressible elastic problem can be defined as:  
\[\
abla\;\sigma+f=0\;in\;\Omega, \tag{26}\]  
\[\sigma\;n=\overline{t}\;on\;\Gamma_{t}, \tag{27}\]  
\[u=\overline{u}\;on\;\Gamma_{u}, \tag{28}\]  
Figure 8: Adaptive mesh required for 1% target error (L\({}_{2}\)-Norm) in compressible elastic square plate with circular inclusion example solution variables considering displacement and stress recovery techniques (Initial regular quadratic triangular mesh: N = 124, DOF = 558). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Displacement Error Recovery**} & \multicolumn{3}{c}{**Stress Error Recovery**} \\ \cline{2-10}
**Recovery Type/Element Type** & **FEM Error** & **Error** & \multicolumn{2}{c}{**Adaptive Mesh**} & **FEM Error** & \multicolumn{2}{c}{**Error**} & \multicolumn{2}{c}{**Adaptive Mesh**} \\ \cline{3-10}  & **Error** & **N** & **DOF** & **Error** & **N** & **DOF** \\ \hline L\({}_{2}\)-Norm-Stress (initial linear triangular mesh: N = 548, DOF = 618, 3% target error) & 7.03 & 6.56 & 667 & 742 & 7.03 & 8.49 & 846 & 934 \\ L\({}_{2}\)-Norm-Stress (initial quadratic triangular mesh: N = 124, DOF = 558, 1% target error) & 3.44 & 6.07 & 231 & 1012 & 3.44 & 3.34 & 183 & 812 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: 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. Figure 7: Adaptive mesh required for 3% target error (L\({}_{2}\)-Norm) in compressible elastic square plate with circular inclusion example solution variables considering displacement and stress recovery techniques (Initial regular linear triangular mesh: N = 548, DOF = 618). The constitutive equation, stress (\(\sigma\)) and strain (\(\varepsilon\)) relation, is written as:  
\[\sigma\;(u)=2\mu\varepsilon(u)+\lambda\;tr\;[\varepsilon(u)]I,\;\text{or} \tag{29}\]  
\[\sigma\;(u,\;p)=2\mu\varepsilon+p\;I,\;\text{or} \tag{30}\]  
\[\frac{p}{\lambda}=div\;(u), \tag{31}\]  
where \(\Omega\) is an incompressible elastic body bounded domain, \(u\) is the displacement field, \(p\) is the pressure, and \(\lambda\), \(\mu\) are the Lame constants. Lame constants are defined as:  
\[\lambda=E/2(I+\
u);\qquad\mu=E\cdot\
u/\left[\left(1-2\cdot\
u\right)\left(1+ \
u\right)\right]\]  
in which \(\overline{t}\) and \(\overline{u}\) are given tractions and displacements on \(\Gamma_{t}\), \(\Gamma_{u}\) boundaries respectively. The two-dimensional incompressible elastic problem subjected to body forces with known solution is implemented to investigate the effectiveness and performance of recovery techniques in terms of convergence properties and required precision adaptive meshes. The domain discretization with triangular elements and node patches for error recovery are shown in Figures 1 and 2. The example was taken by [PERSON] and [PERSON] [26] for recovery technique effectiveness demonstration.