## 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 P_{k1} and P_{k2}, (k_{1} < k_{2}). 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.

Original language | English |
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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.

## Keywords

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