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, (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 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 “Pk1 is more accurate than Pk2 ” becomes greater than 0.5. In these cases, finite element Pk2 is more likely overqualified.
| Original language | English |
|---|---|
| Article number | 84 |
| Journal | Axioms |
| Volume | 11 |
| Issue number | 3 |
| DOIs | |
| State | Published - Mar 2022 |
| Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2022 by the authors. Licensee MDPI, Basel, Switzerland.
Funding
The authors want to warmly dedicate this research to pay homage to the memory of André Avez and Gérard Tronel who largely promote the passion of research and teaching in mathematics.
Keywords
- bramble-hilbert lemma
- error estimates
- finite elements
- probabilistic numerics