Generalized Beta Prime Distribution Applied to Finite Element Error Approximation

In this paper we propose a new generation of probability laws based on the generalized Beta prime distribution to estimate the relative accuracy between two Lagrange finite elements $P_{k_1}$ and $P_{k_2}, (k_1<k_2)$. Since the relative finite element accuracy is usually based on the comparison of the asymptotic speed of convergence when the mesh size $h$ goes to zero, this probability laws highlight that there exists, depending on $h$, cases such that $P_{k_1}$ finite element is more likely accurate than the $P_{k_2}$ one. To confirm this feature, we show and examine on practical examples, the quality of the fit between the statistical frequencies and the corresponding probabilities determined by the probability law. Among others, it validates, when $h$ moves away from zero, that finite element $P_{k_1}$ may produces more precise results than a finite element $P_{k_2}$ since the probability of the event "$P_{k_1}$ is more accurate than $P_{k_2}$" consequently increases to become greater than 0.5. In these cases, $P_{k_2}$ finite elements are more likely overqualified.