On the best accuracy using the $h$-adaptive finite element refinement

In \cite{liu2022practical}, a general algorithm is developed to efficiently obtain the best accuracy using the regular refinement. The adaptive refinement allows for obtaining an accuracy with a smaller number of DoFs compared with the regular refinement. In this paper, we investigate the best accuracy when using the adaptive refinement. To this end, we study the evolution of the truncation error and the round-off error using the adaptive refinement. For the former, a new threshold for the selection of the number of elements to be refined is proposed. For the latter, the round-off error is quantified using the method proposed in \cite{liu2022practical}. Moreover, for achieving a tolerance, we propose to use the line of the round-off error as a stopping criterion.