A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations
A probabilistic approach is developed for the exact solution <i>u</i> to a deterministic partial differential equation as well as for its associated approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semanti...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-12-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/10/4/349 |
| _version_ | 1797506612044234752 |
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| author | Joël Chaskalovic |
| author_facet | Joël Chaskalovic |
| author_sort | Joël Chaskalovic |
| collection | DOAJ |
| description | A probabilistic approach is developed for the exact solution <i>u</i> to a deterministic partial differential equation as well as for its associated approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> performed by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>k</mi></msub></semantics></math></inline-formula> Lagrange finite element. Two limitations motivated our approach: On the one hand, the inability to determine the exact solution <i>u</i> relative to a given partial differential equation (which initially motivates one to approximating it) and, on the other hand, the existence of uncertainties associated with the numerical approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula>. We, thus, fill this knowledge gap by considering the exact solution <i>u</i> together with its corresponding approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> as random variables. By a method of consequence, any function where <i>u</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mrow><mi>h</mi></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> are involved are modeled as random variables as well. In this paper, we focus our analysis on a variational formulation defined on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>W</mi><mrow><mi>m</mi><mo>,</mo><mi>p</mi></mrow></msup></semantics></math></inline-formula> Sobolev spaces and the corresponding a priori estimates of the exact solution <i>u</i> and its approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> in order to consider their respective <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>W</mi><mrow><mi>m</mi><mo>,</mo><mi>p</mi></mrow></msup></mrow></semantics></math></inline-formula>-norm as a random variable, as well as the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>W</mi><mrow><mi>m</mi><mo>,</mo><mi>p</mi></mrow></msup></semantics></math></inline-formula> approximation error with regards to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>k</mi></msub></semantics></math></inline-formula> finite elements. This will enable us to derive a new probability 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 stretchy="false">(</mo><msub><mi>k</mi><mn>1</mn></msub><mo><</mo><msub><mi>k</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T04:35:02Z |
| format | Article |
| id | doaj.art-b8ed70871c8241bc9f9ef4ea7f038be5 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T04:35:02Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-b8ed70871c8241bc9f9ef4ea7f038be52023-11-23T03:50:28ZengMDPI AGAxioms2075-16802021-12-0110434910.3390/axioms10040349A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element ApproximationsJoël Chaskalovic0Jean Le Rond d’Alembert Institute, Sorbonne University, Place Jussieu, CEDEX 05, 75252 Paris, FranceA probabilistic approach is developed for the exact solution <i>u</i> to a deterministic partial differential equation as well as for its associated approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> performed by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>k</mi></msub></semantics></math></inline-formula> Lagrange finite element. Two limitations motivated our approach: On the one hand, the inability to determine the exact solution <i>u</i> relative to a given partial differential equation (which initially motivates one to approximating it) and, on the other hand, the existence of uncertainties associated with the numerical approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula>. We, thus, fill this knowledge gap by considering the exact solution <i>u</i> together with its corresponding approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> as random variables. By a method of consequence, any function where <i>u</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mrow><mi>h</mi></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> are involved are modeled as random variables as well. In this paper, we focus our analysis on a variational formulation defined on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>W</mi><mrow><mi>m</mi><mo>,</mo><mi>p</mi></mrow></msup></semantics></math></inline-formula> Sobolev spaces and the corresponding a priori estimates of the exact solution <i>u</i> and its approximation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>u</mi><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> in order to consider their respective <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>W</mi><mrow><mi>m</mi><mo>,</mo><mi>p</mi></mrow></msup></mrow></semantics></math></inline-formula>-norm as a random variable, as well as the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>W</mi><mrow><mi>m</mi><mo>,</mo><mi>p</mi></mrow></msup></semantics></math></inline-formula> approximation error with regards to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>k</mi></msub></semantics></math></inline-formula> finite elements. This will enable us to derive a new probability 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 stretchy="false">(</mo><msub><mi>k</mi><mn>1</mn></msub><mo><</mo><msub><mi>k</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2075-1680/10/4/349error estimatesfinite elementsBramble–Hilbert lemmaSobolev spaces |
| spellingShingle | Joël Chaskalovic A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations Axioms error estimates finite elements Bramble–Hilbert lemma Sobolev spaces |
| title | A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations |
| title_full | A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations |
| title_fullStr | A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations |
| title_full_unstemmed | A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations |
| title_short | A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations |
| title_sort | probabilistic approach for solutions of deterministic pde s as well as their finite element approximations |
| topic | error estimates finite elements Bramble–Hilbert lemma Sobolev spaces |
| url | https://www.mdpi.com/2075-1680/10/4/349 |
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