# Finite element discretizations of nonlocal minimal graphs: convergence

## TL;DR

This paper develops and analyzes a finite element method for computing fractional minimal graphs, proving convergence and introducing a new geometric error measure, with numerical experiments highlighting the stickiness phenomenon.

## Contribution

It introduces a convergent finite element discretization for fractional minimal graphs and a novel geometric error metric related to normal vectors.

## Key findings

- Convergence in $W^{2r}_1(
abla)$ for all $r<s$.
- A new geometric error measure that captures normal vector discrepancies.
- Numerical experiments demonstrate the stickiness phenomenon.

## 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 $W^{2r}_1(\Omega)$ for all $r<s$, where $W^{2s}_1(\Omega)$ 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.