Explicit k-dependence for Pk finite elements in Wm, p error estimates: application to probabilistic laws for accuracy analysis. Issue 13 (3rd October 2021)

• Record Type: Journal Article
• Authors: Chaskalovic, Joël; Assous, Franck
• Is Part Of: Applicable analysis. Volume 100:Issue 13(2021)
• Page Start: 2825
• Page End: 2843
• Publication Date: 2021-10-03
• Subjects: Error estimates -- finite elements -- Céa lemma -- Bramble–Hilbert lemma -- Banach Sobolev spaces -- probabilistic laws; 65N15 -- 65N75 -- 65N30; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2019.1698727
• Languages: English
• ISSNs: 0003-6811

Abstract

Abstract : We derive an explicit k -dependence in W m, p error estimates for P k Lagrange finite elements. Two laws of probability are established to measure the relative accuracy between P k 1 and P k 2 finite elements, ( k 1 < k 2 ), in terms of W m, p -norms. We further prove a weak asymptotic relation in D ′ ( R ) between these probabilistic laws when difference k 2 − k 1 goes to infinity. Moreover, as expected, one finds that P k 2 finite element is surely more accurate than P k 1, for sufficiently small values of the mesh size h . Nevertheless, our results also highlight cases where P k 1 is more likely accurate than P k 2, for a range of values of h . Hence, this approach brings a new perspective on how to compare two finite elements, which is not limited to the rate of convergence.