Optimal convergence behavior of adaptive FEM driven by simple (h-h/2)-type error estimators

TL;DR

This paper proves that a simple adaptive finite element method using (h-h/2)-type error estimators achieves optimal convergence rates and provides guaranteed lower bounds for the energy error in Poisson problems.

Contribution

It demonstrates the optimal convergence behavior of a straightforward adaptive FEM approach driven by (h-h/2)-type estimators, including guaranteed error bounds.

Findings

Adaptive FEM with (h-h/2)-estimators converges optimally.

Estimators provide guaranteed lower bounds for energy error.

Method is simple to implement and effective.

Abstract

For some Poisson-type model problem, we prove that adaptive FEM driven by the (h-h/2)-type error estimators from [Ferraz-Leite, Ortner, Praetorius, Numer. Math. 116 (2010)] leads to convergence with optimal algebraic convergence rates. Besides the implementational simplicity, another striking feature of these estimators is that they can provide guaranteed lower bounds for the energy error with known efficiency constant 1.