# 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.

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