Patch recovery based on superconvergent derivatives and equilibrium

Abstract

SUMMARYIn this paper a postprocessing technique is developed for determining first-order derivatives (fluxes, stresses) at nodal points based on derivatives in superconvergent points. It is an extension of the superconvergent patch recovery technique presented by Zienkiewicz and Zhu. In contrast to that technique all flux or stress components are interpolated at the same time, coupled by equilibrium equations at the superconvergent points. The equilibrium equations and use of one order higher degree of interpolation polynomials of stress give a dramatic decrease in error of recovered derivatives even at boundaries.

A recovery-explicit error estimator in energy norm for linear elasticity

“…In general they couple the stress components in order to be able to add constraints that improve the quality of the recovered field. Wiberg and Abdulwahab [45,25] proposed to take into account the equilibrium of the recovered field by using a penalty method, Blacker and Belytschko [26] introduced the "Conjoint Polynomial Enhancement" to improve the recovered field along the boundaries. Other techniques looking for equilibrated recovered solutions for upper bounding purposes can be found in [46,47,48,49], but always presenting small lacks of equilibrium even at patch level, thus preventing the strict upper bound property.…”

A separated representation of an error indicator for the mesh refinement process under the proper generalized decomposition framework

“…Generally, it is well known that the main drawback of recovery type techniques is the lack of accuracy along the boundaries of the domain. Further improvements to the SPR technique have been introduced [24][25][26][27] in order to prevent the lack of accuracy of the smoothed field along the boundaries of the domain and also to increase the accuracy into the domain.…”

Validation of A-Posteriori Error Estimators by Numerical Approach

“…Error estimators based on recovery methods are given in [33][34][35][36][37][38][39][40][41][42][43][44]. Here we focus on the superconvergent patch-recovery technique [40][41][42][43][44].…”

“…Here we focus on the superconvergent patch-recovery technique [40][41][42][43][44]. The element error-indicators for elements of any degree p are …”

“…It is possible to modify the definition of the estimators to include the information given by the data and the differential equation. In [42] a modified patch-projection Fx was defined to take into account the additional information, namely…”