# Nodal Accuracy Improvement Technique for Linear Elements with Application to Adaptivity
## 3 Nodal Accuracy Improvement Technology for Linear Elements
### Nodal Accuracy Improvement for 2D and 3D Problems  
For multidimensional problems, the implementation of the nodal accuracy improvement process, as shown in Figure 4, faces a dilemma of choosing between the D-by-D approach and the direct approach, which are equivalent in the FE discretization phase but  
Figure 3: Element D-by-D recovery of 2D problems. only the D-by-D approach is feasible for EEP recovery calculations in the postprocessing phase. This subsection presents a study of this issue and makes a reasonable choice. To avoid unnecessary complications, 2D problems are to be the working case in the following, and the conclusion for 3D problems follows in a similar fashion. The two approaches are briefly described as follows. 1. D-by-D nodal accuracy improvement approach In the D-by-D approach, like the EEP recovery process, the nodal accuracy improvement is carried out at two stages for 2D problems (and three stages for 3D problems) as shown in Figure 4, and can be briefly summarized schematically as follows: * (a) After the line recovery has finished (c.f. Figure 1 and Section 2.2), and before the element recovery, a supplementary problem similar to Equation (11) can be derived for the ODEs of FEMOL (c.f. Equation (8)) and is then solved by using the same FE mesh and trial space used in solving Equation (8) for an improved nodal solution and better FE and EEP solutions on the \"mesh lines\". * (b) After element recovery has finished (c.f. Figure 1 and Section 2.2), another supplementary problem can be derived for the PDE in Equation (1). As before, this additional problem will be solved for using the same FE mesh and trial space used when solving problem (1) for an improved nodal solution for the original PDE problem (1). 2. Direct nodal accuracy improvement approach In the direct approach, skipping the first stage (i.e., step (a)) in the D-by-D approach, the nodal accuracy improvement process is carried out with one stroke for both 2D and 3D problems and is described as follows. After the standard D-by-D EEP calculation has finished for the PDE problem (1), a supplementary problem can be derived for the PDE problem (1) as follows:  
\[\left\{ \begin{array}{ll}L\mathrm{e}^{*}=f-Lu^{*}&\mathrm{in}&\Omega\\ \mathrm{e}^{*}=0&\mathrm{on}&\Gamma_{D}\\ \partial\mathrm{e}^{*}/\partial\mathrm{n}=0&\mathrm{on}&\Gamma_{N}\end{array} \right. \tag{13}\]  
which is solved by using the same FE mesh and trial space as used in solving problem (1) and hence is reduced to a back-substitution process similar to Equation (12). It is seen that the direct approach is very simple and straightforward and the supplementary problem in Equation (13) is also directly applicable to 3D problems. Figure 4: Two approaches for nodal accuracy improvement for 2D problems. In order to demonstrate the improvement effect of nodal accuracy intuitively, define a test problem on an irregular domain \(\Omega\) as shown in Figure 4a with homogeneous Dirichlet boundary condition on all boundaries, and set the exact solution to be  
\[\left\{ \begin{array}{l}u=\cos(\frac{\pi\,a}{2})\cos(\frac{\pi\,\beta}{2})\\ \alpha=\frac{y}{3}-5+\frac{\pi}{6}+\frac{\sqrt{4y^{2}-120y+4 xy+576+48x+x^{2}}} {9}\\ \beta=\frac{x}{9}+\frac{2y}{9}+\frac{5}{3}-\frac{\sqrt{4y^{2}-120y+4 xy+576+48x+ x^{2}}}{9}\end{array} \right. \tag{14}\]  
Using uniform mesh of \(2\times 2\) (Figure 4c), \(4\times 4\), \(8\times 8\) and \(16\times 16\) elements, the maximum errors of nodal displacements among all the nodes are calculated for the FE solution, and the improved solutions of both approaches are shown in Figure 5, where the element size parameter \(h\) is taken to be the bottom side length of the bottom elements. Note that all of the computations are conducted on the same mesh by using the same linear elements. It is seen from Figure 5 that for linear elements, nodal displacements of FEM only achieve convergence \(O(h^{2})\).