Finite element algorithms for nonlocal minimal graphs

TL;DR

This paper develops finite element algorithms to compute fractional minimal graphs, addressing both computational methods and qualitative properties of solutions for nonlocal minimal surface problems.

Contribution

It introduces finite element discretization, gradient flow, and Newton schemes for fractional minimal graphs, with analysis of Dirichlet data truncation effects.

Findings

Effective finite element algorithms for fractional minimal graphs.

Numerical experiments illustrating qualitative features.

Quantification of Dirichlet data truncation impact.

Abstract

We discuss computational and qualitative aspects of the fractional Plateau and the prescribed fractional mean curvature problems on bounded domains subject to exterior data being a subgraph. We recast these problems in terms of energy minimization, and we discretize the latter with piecewise linear finite elements. For the computation of the discrete solutions, we propose and study a gradient flow and a Newton scheme, and we quantify the effect of Dirichlet data truncation. We also present a wide variety of numerical experiments that illustrate qualitative and quantitative features of fractional minimal graphs and the associated discrete problems.