Generalized Beta Prime Distribution Applied to Finite Element Error Approximation

In this paper, we propose a new family of probability laws based on the Generalized Beta Prime distribution to evaluate the relative accuracy between two Lagrange finite elements P_k1 and P_k2, (k₁ < k₂). Usually, the relative finite element accuracy is based on the comparison of the asymptotic speed of convergence, when the mesh size h goes to zero. The new probability laws we propose here highlight that there exists, depending on h, cases where the P_k1 finite element is more likely accurate than the P_k2 element. To confirm this assertion, we highlight, using numerical examples, the quality of the fit between the statistical frequencies and the corresponding probabilities, as determined by the probability law. This illustrates that, when h goes away from zero, a finite element P_k1 may produce more precise results than a finite element P_k2, since the probability of the event “P_k1 is more accurate than P_k2 ” becomes greater than 0.5. In these cases, finite element P_k2 is more likely overqualified.