Finite element algorithms for nonlocal minimal graphs

Juan Pablo Borthagaray; Wenbo Li; Ricardo H. Nochetto; Juan Pablo Borthagaray; Wenbo Li; Ricardo H. Nochetto

doi:10.3934/mine.2022016

Mathematics in Engineering

2022, Volume 4, Issue 2: 1-29. doi: 10.3934/mine.2022016

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Research article

Finite element algorithms for nonlocal minimal graphs

1.
Departamento de Matemática y Estadística del Litoral, Universidad de la República, Salto, Uruguay
2.
Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA
3.
Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA

Received: 26 May 2021 Accepted: 07 July 2021 Published: 20 July 2021

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.
- nonlocal minimal surfaces,
- finite elements,
- fractional diffusion
Citation: Juan Pablo Borthagaray, Wenbo Li, Ricardo H. Nochetto. Finite element algorithms for nonlocal minimal graphs[J]. Mathematics in Engineering, 2022, 4(2): 1-29. doi: 10.3934/mine.2022016

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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.

References

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