The superconvergent patch recovery and <i>a posteriori</i> error estimates. Part 2: Error estimates and adaptivity

TLDR

This paper discusses a posteriori error estimation in the second part of the study. The authors derive a theorem linking the effectivity index of the Zienkiewicz–Zhu error estimator to the convergence rate of the recovered solution. The study employs the superconvergent recovery method from part 1 to compute the Zienkiewicz–Zhu error estimator and accurately estimate the exact error. The results demonstrate that superconvergent recovery drives the effectivity index toward one, with numerical tests confirming excellent effectivity in energy and gradient norms and showing strong performance in adaptive mesh refinement.

Abstract

Abstract In this second part of the paper, the issue of a posteriori error estimation is discussed. In particular, we derive a theorem showing the dependence of the effectivity index for the Zienkiewicz–Zhu error estimator on the convergence rate of the recovered solution. This shows that with superconvergent recovery the effectivity index tends asymptotically to unity. The superconvergent recovery technique developed in the first part of the paper 1 is the used in the computation of the Zienkiewicz–Zhu error estimator to demonstrate accurate estimation of the exact error attainable. Numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient (stress) error estimation. Several examples of the performance of the error estimator in adaptive mesh refinement are also presented.