Year: 2024
Type: article
Abstract: We consider finite element approximations to the optimal constant for the Hardy inequality with exponent p=2 p = 2 in bounded domains of dimension n=1 n = 1 or n\geq 3 n ≥ 3 . For finite element spaces of piecewise linear and continuous functions on a mesh of size h h , we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to 1/|\log h|^2 1 / ∣ log h ∣ 2 . This result holds in dimension n=1 n = 1 , in any dimension n\geq 3 n ≥ 3 if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension n=3 n = 3 for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally.
Source: Journal of convex analysis
DOI: 10.68381/jca31027
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Funders: Alexander von Humboldt-Stiftung, Università degli Studi di Napoli Federico II, Compagnia di San Paolo, Istituto Nazionale di Alta Matematica "Francesco Severi", Ministerul Cercetării, Inovării şi Digitalizării, Friedrich-Alexander-Universität Erlangen-Nürnberg, Deutsche Forschungsgemeinschaft, Ministero dell’Istruzione, dell’Università e della Ricerca, Academia Româna, Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii, Universitatea din București, Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni
OpenAlex ID: W4385895100