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Fundamentals of the finite element method
Published in Paulo B. Lourenço, Angelo Gaetani, Finite Element Analysis for Building Assessment, 2022
Paulo B. Lourenço, Angelo Gaetani
As superconvergent points provide a higher order of accuracy, it is possible to improve the overall accuracy of the finite element solution after the analysis. This process is referred to as recovery of strains and stresses. Various recovery methods are described in literature. Here, an example is given considering the recovered stresses obtained by interpolating nodal values the same way done for displacements, see Eq. (1.90). Nodal values of strains εnodal and stresses σnodal can be calculated by a two-step procedure: (1) extrapolating them from the values at the integration points and (2) averaging these values in the case the node is shared between two or more elements. Figure 1.30 schematically illustrates the method providing a good estimation of the nodal value. It is worth noticing that two Gauss integration points can be interpolated only with a straight line. Note also that averaging in Figure 1.28b provides an excellent estimation. ε≈Nεnodalσ≈Nσnodal
Introduction to the Finite Element Method
Published in Victor N. Kaliakin, Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods, 2018
In Figures 6.8 and 6.10 the finite element solution is seen to be exact at the nodes; that is, φ^(x1)=φ(xi) for i = 1,2, ... , P. This result holds for all Galerkin finite element solutions in one-dimension. A formal proof of this result is given by Hughes [252], Such exceptional accuracy characteristics are often referred to as superconvergence phenomena. They occur only in rather simple problems; in more complicated problems, superconvergence cannot be guaranteed.
Superconvergence analysis for a semilinear parabolic equation with BDF-3 finite element method
Published in Applicable Analysis, 2022
On the other hand, superconvergence of FEMs is a phenomenon that the convergence rate exceeds what general cases can provide, which can be achieved for smoother solutions with structured meshes. Many studies have been conducted for superconvergence analysis; see, e.g. [15–17]. In fact, in order to improve the approximation accuracy of the finite element solution, the idea of high precision has become a main research direction of FEM. As early as the 1960s, superconvergence of finite element solutions or their derivatives have been found in engineering. In the theoretical analysis and practical calculation, if there are good meshes (such as almost uniform meshes), the error between finite element solution and the finite element interpolation of the true solution is much smaller than that between finite element solution and true solution in a sense of norm, which is also called superclose [18,19]. At the same time, on the premise of obtaining the superclose property, the global superconvergence result [20–22] can be obtained by interpolating the finite element solution.