Finite element discretizations of nonlocal minimal graphs: Convergence

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 W12r(Ω) for all r<s, where W12s(Ω) is closely related to the natural energy space. Moreover, we introduce a geometric notion of error that, for any pair of H1 functions, in the limit s→1∕2 recovers a weighted L2-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.