Comparison of some a posteriori error estimators

Abstract

The idea of {\it a posteriori } error estimates based on the reconstruction of the equilibrated potential and/or equilibrated flux goes back to the Prager-Synge equality %\cite{Pra}
for the Poisson equation $ -\Delta p = f $. This identity is valid for all $ v \in H^1_0(\Omega) $ and all
$ \mathbf{u} \in H (\Div, \Omega) $ such that $ \Div \mathbf{u} + f = 0 $, and given by
\[
\|\mathbf{u}-\nabla v\|^2_{0,\Omega}=\| \mathbf{u}-\nabla p\|^2_{0,\Omega}+\|\nabla p-\nabla v\|^2_{0,\Omega}.
\]
It follows that, to obtain such estimate, we need to reconstruct a so-called equilibrated flux; $ \mathbf{u} \in H (\Div; \Omega) $ satisfying the equilibrium condition $ \Div \mathbf{u} + f = 0 $ and such that $ \vu- \nabla p $ is as small as possible, and/or reconstruct a potential $ v $ in $ H^1_0 (\Omega) $.\\
In all cases, to have an estimate, which is said "by reconstruction", it is necessary to have at the end an equilibrated, flux and potential. Now, the question is: is it better to work with a numerical method that allows us to have an equilibrated quantities and in this case there is no need to reconstruct, or else, do we use a method where, we do not have an equilibrated solutions such as the Discontinuous Galerkin method, and in this case it is necessary to reconstruct the two variables?
We first compare two types of error estimators: the classical residual-based estimators, which do not require any reconstruction, and the reconstruction-based estimators, in the context of a diffusion problem. Then, using various numerical approximation methods, we proceed to compare the different reconstruction-based estimators.