Generalized Beta Prime Distribution Applied to Finite Element Error Approximation

TL;DR

This paper introduces a new probabilistic approach using the generalized Beta prime distribution to compare the accuracy of finite element methods, revealing scenarios where lower-order elements may outperform higher-order ones depending on mesh size.

Contribution

It develops a novel probability law for finite element accuracy comparison, validated through practical examples, highlighting non-asymptotic accuracy relationships.

Findings

Finite element accuracy can favor lower-order elements at certain mesh sizes.

The proposed probability law fits well with empirical frequency data.

Higher-order elements are not always more accurate than lower-order ones.

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 $P_{k_1}$ 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.