# Nodal Accuracy Improvement Technique for Linear Elements with Application to Adaptivity
## 5 Numerical Examples
### Singularity Problem  
The governing equation of the problem is defined in an L-shaped domain \(\Omega\) shown in Figure 19 as follows:  
\[\left\{\begin{array}{ll}-\
abla^{2}u=0&\text{in }\Omega\\ u=0&\text{on }y=-5,\,0\leq x\leq 5\\ u=1&\text{on }y=5,\,-5\leq x\leq 5\\ \partial u/\partial n=0&\text{on the rest boundary}\end{array}\right. \tag{26}\]  
The \"exact solution\" was obtained on a high-density mesh, and the effective gradient is plotted in Figure 20. The fact is that \(\partial u/\partial r\) exhibits a singularity of \((1/r)^{1/3}\) at the origin [30], where \(r\) denotes the distance from the origin. Taking Figure 19 as the initial mesh of three linear elements and setting \(T_{I}=5\times 10^{-3}\), the mesh \(\pi^{*}\) from the first phase is obtained after five adaptivity steps as shown in Figure 21a. The final mesh \(\pi\), as shown in Figure 21b, is obtained after three more adaptivity steps in the second phase of adaptivity.  
Similar to Section 5.3, the FE solution on mesh \(\pi^{*}\) does not meet the ultimate adaptivity criterion set in Equation (18). It is after the follow-up checking and further refinement locally in the second phase that the FE solution on the final mesh \(\pi\), generated by satisfying the stopping criterion in Equation (19), becomes fully satisfactory with Equation (18), and the contours of the distribution of the estimated and real pointwise errors on the final mesh \(\pi\) are presented in Figure 22.  
Setting \(T_{I}=5\times 10^{-3}\), \(N_{\text{dof}}\) is only 240 from the adaptive refinement, while \(N_{\text{dof}}\) is 12,412 from the uniform refinement. Giving different \(T_{I}\), the convergence rates of the three refinement schemes are calculated as shown in Figure 23. It is noted that, for linear elements, the adaptive convergence rate tends to be the optimal \((m+1)/2=1\), while the uniform convergence rate is dominated by the singularity index \(1/3\), which is very slow  
Figure 19: L-shaped domain and initial mesh.  
Figure 18: Convergence rates of uniform refinement and adaptive refinement.  
In short, the first three examples show that improved EEP displacements gain super-convergence compared with traditional FEM and EEP solutions. Subsequent adaptive examples show that the adaptive solution based on EEP is extended to linear elements successfully by the self-adaptive algorithm proposed in this paper with the following conclusions. Firstly, the adaptive linear FE solution satisfies the user-specified error tolerances in maximum norm on the final meshes well. Secondly, the adaptive local mesh refinement provides nearly optimal convergence rates and tremendously reduces the total cost of DOFs. Thirdly, for the stress concentration problem, the adaptivity efficiency of linear elements is not as efficient as that of cubic elements for stringent tolerances, but for relatively loose tolerances both are roughly at par. Finally, for the singularity problem, the adaptivity efficiency of linear elements even exceeds cubic elements for a wide range of the mostly used tolerances.