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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><sem...
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| Format: | Article |
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MDPI AG
2022-02-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/11/3/84 |
| _version_ | 1797472865540374528 |
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| author | Joël Chaskalovic Franck Assous |
| author_facet | Joël Chaskalovic Franck Assous |
| author_sort | Joël Chaskalovic |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>1</mn></msub></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub><mo>,</mo><mrow><mo>(</mo><msub><mi>k</mi><mn>1</mn></msub><mo><</mo><msub><mi>k</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Usually, the relative finite element accuracy is based on the comparison of the asymptotic speed of convergence, when the mesh size <i>h</i> goes to zero. The new probability laws we propose here highlight that there exists, depending on <i>h</i>, cases where the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>1</mn></msub></msub></semantics></math></inline-formula> finite element is more likely accurate than the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub></semantics></math></inline-formula> 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 <i>h</i> goes away from zero, a finite element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>1</mn></msub></msub></semantics></math></inline-formula> may produce more precise results than a finite element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub></semantics></math></inline-formula>, since the probability of the event “<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>1</mn></msub></msub></semantics></math></inline-formula><i>is more accurate than</i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub></semantics></math></inline-formula>” becomes greater than 0.5. In these cases, finite element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub></semantics></math></inline-formula> is more likely overqualified. |
| first_indexed | 2024-03-09T20:07:12Z |
| format | Article |
| id | doaj.art-ddad0b7bc224433d857e148c1b56bc1a |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T20:07:12Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-ddad0b7bc224433d857e148c1b56bc1a2023-11-24T00:28:35ZengMDPI AGAxioms2075-16802022-02-011138410.3390/axioms11030084Generalized Beta Prime Distribution Applied to Finite Element Error ApproximationJoël Chaskalovic0Franck Assous1Jean Le Rond d’Alembert Institute, Sorbonne University, 4 Place Jussieu, CEDEX 05, 75252 Paris, FranceDepartment of Mathematics, Ariel University, Ariel 40700, IsraelIn 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>1</mn></msub></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub><mo>,</mo><mrow><mo>(</mo><msub><mi>k</mi><mn>1</mn></msub><mo><</mo><msub><mi>k</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Usually, the relative finite element accuracy is based on the comparison of the asymptotic speed of convergence, when the mesh size <i>h</i> goes to zero. The new probability laws we propose here highlight that there exists, depending on <i>h</i>, cases where the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>1</mn></msub></msub></semantics></math></inline-formula> finite element is more likely accurate than the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub></semantics></math></inline-formula> 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 <i>h</i> goes away from zero, a finite element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>1</mn></msub></msub></semantics></math></inline-formula> may produce more precise results than a finite element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub></semantics></math></inline-formula>, since the probability of the event “<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>1</mn></msub></msub></semantics></math></inline-formula><i>is more accurate than</i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub></semantics></math></inline-formula>” becomes greater than 0.5. In these cases, finite element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>k</mi><mn>2</mn></msub></msub></semantics></math></inline-formula> is more likely overqualified.https://www.mdpi.com/2075-1680/11/3/84error estimatesfinite elementsbramble-hilbert lemmaprobabilistic numerics |
| spellingShingle | Joël Chaskalovic Franck Assous Generalized Beta Prime Distribution Applied to Finite Element Error Approximation Axioms error estimates finite elements bramble-hilbert lemma probabilistic numerics |
| title | Generalized Beta Prime Distribution Applied to Finite Element Error Approximation |
| title_full | Generalized Beta Prime Distribution Applied to Finite Element Error Approximation |
| title_fullStr | Generalized Beta Prime Distribution Applied to Finite Element Error Approximation |
| title_full_unstemmed | Generalized Beta Prime Distribution Applied to Finite Element Error Approximation |
| title_short | Generalized Beta Prime Distribution Applied to Finite Element Error Approximation |
| title_sort | generalized beta prime distribution applied to finite element error approximation |
| topic | error estimates finite elements bramble-hilbert lemma probabilistic numerics |
| url | https://www.mdpi.com/2075-1680/11/3/84 |
| work_keys_str_mv | AT joelchaskalovic generalizedbetaprimedistributionappliedtofiniteelementerrorapproximation AT franckassous generalizedbetaprimedistributionappliedtofiniteelementerrorapproximation |