Generalized Beta Prime Distribution Applied to Finite Element Error Approximation
Abstract
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 Pk1 and Pk2, (k1<k2). 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 Pk1 finite element is more likely accurate than the Pk2 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 Pk1 may produces more precise results than a finite element Pk2 since the probability of the event "Pk1 is more accurate than Pk2" consequently increases to become greater than 0.5. In these cases, Pk2 finite elements are more likely overqualified.