Generalized Beta Prime Distribution Applied to Finite Element Error Approximation

Abstract

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 Pk1 and Pk2,(k1k2). 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 Pk1 finite element is more likely accurate than the Pk2 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 Pk1 may produce more precise results than a finite element Pk2, since the probability of the event “Pk1is more accurate thanPk2” becomes greater than 0.5. In these cases, finite element Pk2 is more likely overqualified.

Keywords

• error estimates
• finite elements
• bramble-hilbert lemma
• probabilistic numerics