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Constructive Approximation on Graded Meshes for the Integral Fractional Laplacian

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Constructive Approximation Aims and scope

Abstract

We consider the homogeneous Dirichlet problem for the integral fractional Laplacian \((-\Delta )^s\). We prove optimal Sobolev regularity estimates in Lipschitz domains provided the solution is \(C^s\) up to the boundary. We present the construction of graded bisection meshes by a greedy algorithm and derive quasi-optimal convergence rates for approximations to the solution of such a problem by continuous piecewise linear functions. The nonlinear Sobolev scale dictates the relation between regularity and approximability.

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Author notes

  1. Juan Pablo Borthagaray

    Present address: Centro de Matemática, Universidad de la República, Montevideo, Uruguay

Authors and Affiliations

  1. Departamento de Matemática y Estadística del Litoral, Universidad de la República, Salto, Uruguay

    Juan Pablo Borthagaray

  2. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

Authors
  1. Juan Pablo Borthagaray
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  2. Ricardo H. Nochetto
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Correspondence to Juan Pablo Borthagaray.

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Communicated by Wofgang Dahmen.

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JPB has been supported in part by Fondo Vaz Ferreira Grant 2019-068. RHN has been supported in part by NSF Grant DMS-1908267.

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Borthagaray, J.P., Nochetto, R.H. Constructive Approximation on Graded Meshes for the Integral Fractional Laplacian. Constr Approx 57, 463–487 (2023). https://doi.org/10.1007/s00365-023-09617-5

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  • DOI: https://doi.org/10.1007/s00365-023-09617-5

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