Finite element discretizations of nonlocal minimal graphs: Convergence. (December 2019)

• Record Type: Journal Article
• Authors: Borthagaray, Juan Pablo; Li, Wenbo; Nochetto, Ricardo H.
• Is Part Of: Nonlinear analysis. Volume 189(2019)
• Publication Date: 2019-12
• Subjects: 49Q05 -- 35R11 -- 65N12 -- 65N30; Nonlocal minimal surfaces -- Finite elements -- Fractional diffusion; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.06.025
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we propose and analyze a finite element discretization for the computation of fractional minimal graphs of order s ∈ ( 0, 1 ∕ 2 ) on a bounded domain Ω . Such a Plateau problem of order s can be reinterpreted as a Dirichlet problem for a nonlocal, nonlinear, degenerate operator of order s + 1 ∕ 2 . We prove that our numerical scheme converges in W 1 2 r ( Ω ) for all r < s, where W 1 2 s ( Ω ) is closely related to the natural energy space. Moreover, we introduce a geometric notion of error that, for any pair of H 1 functions, in the limit s → 1 ∕ 2 recovers a weighted L 2 -discrepancy between the normal vectors to their graphs. We derive error bounds with respect to this novel geometric quantity as well. In spite of performing approximations with continuous, piecewise linear, Lagrangian finite elements, the so-called stickiness phenomenon becomes apparent in the numerical experiments we present.