# Nodal Accuracy Improvement Technique for Linear Elements with Application to Adaptivity
## 2 Basic Theory
### Element Energy Projection Recovery Strategy  
The basic assumption of the EEP method is that the projection theorem, i.e., \(a(e^{h},v^{h})=0,\ \forall v^{h}\in S^{h}\), holds almost true within each individual element. For 1D problems, it is mathematically proved that the neglected error for the element projection is of \(O(h^{2m})\) using elements of degree \(m\). For multidimensional problems, a semi-discrete method named the finite element method of lines (FEMOL) [27] is used as a bridge between 1D and 2D problems. Taking problem (1) as the model problem, the difference between FEMOL and FEM is that FEMOL is a semi-discrete numerical method, and thus the full discretization in Equation (4) is replaced by a corresponding FEMOL semi-discrete expression  
\[\widetilde{u}^{h}(\xi,\eta)=\sum_{i=1}^{2}N_{i}(\xi)d_{i}(\eta)=\mathbf{N}( \xi)\mathbf{d}(\eta)^{c} \tag{7}\]which, by using energy variational means, leads to the following ordinary differential equations (ODEs) of FEMOL:  
\[\left\{\begin{array}{l}\mathbf{Ld}\equiv-(\mathbf{Ad}^{\prime}+\mathbf{Bd})^{ \prime}+\mathbf{B}^{T}\mathbf{d}^{\prime}+\mathbf{Cd}=\mathbf{F},\;\eta\in(-1, \;1)\\ \text{withBCs}\end{array}\right. \tag{8}\]  
Then, the FEM is used again to solve the above FEMOL ODEs by assuming that  
\[\mathbf{d}(\eta)^{\epsilon}=\sum_{j=1}^{2}N_{j}(\eta)\mathbf{d}_{j}^{ \epsilon} \tag{9}\]  
The two-step discretization process of Equations (7) and (9) is equivalent to the full discretization process of Equation (4), and the same matrix stiffness equation as Equation (6) can be derived. Then, the algorithm for 2D EEP super-convergent solutions, which is also called Global D-by-D recovery strategy (EEP-G), can be achieved by two generalized 1D recovery processes as shown schematically in Figure 1.  
Similar to 2D problems, the finite element method of faces (FEMOF) is used as a bridge between 2D and 3D problems, then the Global D-by-D recovery strategy can be used to calculate super-convergent solutions of 3D problems. Discretization and recovery processes for 3D problems are schematically shown in Figure 2.  
In recent years, more detailed work has been carried out on the super-convergence algorithm of 2D EEP, and an element D-by-D recovery strategy (EEP-E) was proposed [22], which includes four recovery steps: side node recovery, side recovery, internal side recovery and element recovery, as shown in Figure 3.  
Compared with EEP-G, EEP-E gains similar recovery results but is implemented almost on a single element, which leads to less computation and more flexible FE meshes, such as local refinement, for recovery and adaptivity progresses.  
Figure 1: Global D-by-D discretization and recovery of 2D problems.  
Figure 2: Global D-by-D discretization and recovery of 3D problems.  
Let \((\ )^{h}\) and \((\ )^{*}\) denote the FE solution and EEP solution, respectively. Let \(e_{\max}^{*}\) denote the maximum errors of interior EEP displacement \(u^{*}\), i.e., \(e_{\max}^{*}=\underset{\Omega}{\max}|u-u^{*}|\). It is worth noting that the super-convergent solution \(u^{*}\) does not change the FE solution \(u^{h}\) at corner nodes, i.e., \(u_{i}^{h}=u_{i}^{*}\) at element corners, and hence the super-convergence in \(u^{*}\) is achieved based on the super-convergent nodal values. In other words, the super-convergence order of \(u^{*}\) cannot exceed the convergence order of the FE nodal solution \(u_{i}^{h}\). Therefore, although the EEP displacement \(u^{*}\) is super-convergent compared to the FE displacement \(u^{h}\) for elements of degree \(m>1\), the EEP displacement \(u^{*}\) for linear elements can only achieve convergence \(O(h^{2})\) due to its second-order accuracy of nodal displacements.