{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T03:00:26Z","timestamp":1769050826460,"version":"3.49.0"},"reference-count":14,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,12,20]],"date-time":"2021-12-20T00:00:00Z","timestamp":1639958400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A probabilistic approach is developed for the exact solution u to a deterministic partial differential equation as well as for its associated approximation uh(k) performed by Pk Lagrange finite element. Two limitations motivated our approach: On the one hand, the inability to determine the exact solution u 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 uh(k). We, thus, fill this knowledge gap by considering the exact solution u together with its corresponding approximation uh(k) as random variables. By a method of consequence, any function where u and uh(k) are involved are modeled as random variables as well. In this paper, we focus our analysis on a variational formulation defined on Wm,p Sobolev spaces and the corresponding a priori estimates of the exact solution u and its approximation uh(k) in order to consider their respective Wm,p-norm as a random variable, as well as the Wm,p approximation error with regards to Pk finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements Pk1 and Pk2,(k1&lt;k2).<\/jats:p>","DOI":"10.3390\/axioms10040349","type":"journal-article","created":{"date-parts":[[2021,12,20]],"date-time":"2021-12-20T02:50:18Z","timestamp":1639968618000},"page":"349","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Probabilistic Approach for Solutions of Deterministic PDE\u2019s as Well as Their Finite Element Approximations"],"prefix":"10.3390","volume":"10","author":[{"given":"Jo\u00ebl","family":"Chaskalovic","sequence":"first","affiliation":[{"name":"Jean Le Rond d\u2019Alembert Institute, Sorbonne University, Place Jussieu, CEDEX 05, 75252 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1515\/cmam-2018-0270","article-title":"A new probabilistic interpretation of Bramble-Hilbert lemma","volume":"20","author":"Chaskalovic","year":"2019","journal-title":"Comput. Methods Appl. Math."},{"key":"ref_2","unstructured":"Chaskalovic, J., and Assous, F. (2018). A new mixed functional-probabilistic approach for finite element accuracy. arXiv."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Ciarlet, P.G., and Lions, J.L. (1991). Basic error estimates for elliptic problems. Handbook of Numerical Analysis, North Holland.","DOI":"10.1016\/S1570-8659(05)80039-0"},{"key":"ref_4","unstructured":"Raviart, P.A., and Thomas, J.M. (1982). Introduction \u00e0 L\u2019analyse Num\u00e9rique des \u00c9quations aux D\u00e9riv\u00e9es Partielles, Masson."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1137\/0707006","article-title":"Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation","volume":"7","author":"Bramble","year":"1970","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2825","DOI":"10.1080\/00036811.2019.1698727","article-title":"Explicit k-dependenc for Pk finite elements in Wm,p error estimates: Application to probability distributions for accuracy analysis","volume":"100","author":"Chaskalovic","year":"2019","journal-title":"Appl. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Ern, A., and Guermond, J.L. (2004). Theory and Practice of Finite Elements, Springer.","DOI":"10.1007\/978-1-4757-4355-5"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ciarlet, P.G. (2002). The Finite Element Method for Elliptic Problems, SIAM. Classics In Applied Mathematics.","DOI":"10.1137\/1.9780898719208"},{"key":"ref_9","unstructured":"Brezis, H. (1992). Analyse Fonctionnelle\u2014Th\u00e9orie et Applications, Masson."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"515","DOI":"10.3934\/eect.2016017","article-title":"A uniform discrete inf-sup inequality for finite element hydro-elastic models","volume":"5","author":"Toundykov","year":"2016","journal-title":"Evol. Equ. Control Theory"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chaskalovic, J. (2013). Mathematical and Numerical Methods for Partial Differential Equations, Springer.","DOI":"10.1007\/978-3-319-03563-5"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"684","DOI":"10.3846\/mma.2021.14079","article-title":"Numerical validation of probabilistic laws to evaluate finite element error estimates","volume":"26","author":"Chaskalovic","year":"2021","journal-title":"Math. Model. Anal."},{"key":"ref_13","unstructured":"Crouzeix, M., and Mignot, A.L. (1984). Analyse Num\u00e9rique des \u00c9quations Diff\u00e9centielles, Masson."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Stroud, A.H. (1974). Numerical Quadrature and Solution of Ordinary Differential Equations, Springer.","DOI":"10.1007\/978-1-4612-6390-6"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/4\/349\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:52:14Z","timestamp":1760169134000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/4\/349"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,20]]},"references-count":14,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["axioms10040349"],"URL":"https:\/\/doi.org\/10.3390\/axioms10040349","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,20]]}}}