# Nodal Accuracy Improvement Technique for Linear Elements with Application to Adaptivity
## 1 Introduction  
Compared with the traditional finite element method (FEM), the adaptive finite element (FE) method can obtain an optimal FE mesh with its corresponding FE solution satisfying the user-specified error tolerances. The error estimation in the adaptive FE solutions plays a crucial role in the performance of adaptive algorithms in terms of effectiveness, efficiency and reliability. In recent years, various a posteriori error estimators for the FEM have been developed since it was first proposed by [PERSON] and [PERSON] [1]. In general, these methods can fall into two categories [2], i.e., residual-based error estimators and recovery-based error estimators, of which the latter is more robust and popular [3]. For residual-based error estimators, the non-zero residuals of the FE solutions are used in the computed region or on the boundary, either explicitly [1, 4] or implicitly [5]. For recovery-based error estimators, super-convergent solutions are calculated, replacing the exact solutions, to estimate the errors of FE solutions. Many representative recovery methods have been proposed [6, 7, 8, 9, 10, 11, 12, 13, 14], some of these methods rely on natural super-convergent points such as the Gauss points or Lobatto points in each element [6, 7], and others do not use natural super-convergent points directly [8, 9, 10, 11, 12, 13, 14]. Adaptive analyses based on the above a posteriori error estimators have been developed, and in most cases, the energy norm is used to measure and control the calculated errors [15, 16, 17, 18, 19, 20], though it makes less practical from an engineering perspective. The maximum norm is more stringent and straightforward than the energy norm, but for singularity problems it is more difficult to control the errors instresses directly by maximum norm than in displacements, and therefore the maximum norm of displacements is a reasonably more appropriate choice [21; 22], which is the choice and highlight of this paper. It is noted that most recovery-based error estimators focus on achieving better stress solutions [6; 7; 8; 9; 10; 11; 12; 13; 14]. Although some of these methods can also be used for recovering displacements with better accuracy, the recovered displacements for linear elements using the above methods are no longer super-convergent [11; 22; 23]. It seems that the second-order convergence accuracy of interior displacements is the best accuracy that can be achieved for linear elements due to the constraint of second-order accuracy of nodal displacements [24; 25]. It is widely accepted that the error estimations of using super-convergent solutions are reliable theoretically in recovery-based methods. Therefore, the linear elements have long been precluded from the adaptive FEM, especially using maximum norm to control errors of displacements, since both the FE and recovery solutions gain the same convergence order [22]. In recent years, a novel recovery-based technique named the Element Energy Projection (EEP) method was proposed by [PERSON], etc. [11], which achieves point-wise convergence at least one order higher than the corresponding FE solutions for elements of degree \(m\) (\(>\)1) and hence serves as a reliable error estimator for adaptive FE analyses. As a result, the EEP technique has been successfully applied to a series of adaptive FE analyses in the maximum norm for a variety of problems [21; 22; 26]. Again, the most commonly used linear elements have been precluded from both the EEP method and the EEP-based adaptivity analyses [21; 22] for the same reason that the second-ordered nodal accuracy of linear elements for 1D to 3D problems prevents super-convergence from being achieved over the original linear elements. This article aims to overcome the aforementioned difficulties and obstacles for linear elements by proposing a nodal accuracy improvement technique.