The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique

Abstract

SUMMARYThis is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an 0(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L, projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L, projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.

Studies on Barlow points, Gauss points and Superconvergent points in 1D with Lagrangian and Hermitian finite element basis

“…By comparing the zeros of u k and s k (see Table 4 and Table 5) one can conclude that an O(h k+2 ) convergence is impossible to happen mathematically with a straightforward application of the Taylor's Series paradigm. Therefore, the ultraconvergence phenomenon as notified by Zienkiewicz in his numerical experiments are believed to be a numerical coincidence (see reference [13]). It is important to comment the existence of purely imaginary ultraconvergent zeros as one can infer from Table 4.…”

“…The superconvergence is also linked to the concept of reduced integration and zero energy modes so important in the Finite Element Theory [15]. Even more important, the superconvergence points are of utility as sampling points for the finite element designers in formulations like gradient smoothening or gradient recovery in the sense of The Superconvergent Patch Recovery as sketched by Zienkiewcz [13]. However, the major significance of these points that explains all kind of numerical and analytical studies in this topic refers to its application in adaptive finite element analysis, especially for construction of error indicators based on the recovered gradient or hessian.…”

Monte Carlo Assessment of the Impact of Oscillatory and Pulsating Boundary Conditions on the Flow Through Porous Media

“…Regarding accuracy, our numerical experiments showed that this strategy produced sufficiently reliable results. We note that improvements of the accuracy can be obtained using gradient recovery techniques which yield superconverging behaviour (Zienkiewicz and Zhu 1992). The aim of this study is to investigate the impact of oscillatory and pulsating boundary conditions on the volume flow rate at the right end of the tube.…”

The Scaled Boundary Finite Element Method for the Efficient Modelling of Linear Elastic Fracture

“…Using a stress recovery technique such as the Super-convergent Patch Recovery (SPR) theory ( [29], [30]), the raw stress field may be recovered resulting in significantly improved results. Next to the fact that only the singular modes have to be smoothed on the boundary, numerical experiments have shown that excellent results can still be achieved by only recovering the stresses over a few element in the general crack extension direction, further reducing computational cost.…”