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~$s\in (0,1/2)$ on a bounded domain $\Omega$. 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 $W_1^{2r}(\Omega )$ for all $r<s$, where $W_1^{2s}(\Omega )$ 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 $s\to 1/2$ 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 {\em stickiness} phenomenon becomes apparent in the numerical experiments we present.

Lemma 4.1 was removed due to a mistake in its proof. This only affects the new Theorem 4.2 (convergence), whose proof has been corrected