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Regularity theory and high order numerical methods for the (1D)-fractional Laplacian

Acosta, Gabriel and Borthagaray, Juan Pablo and Bruno, Oscar and Maas, Martín (2018) Regularity theory and high order numerical methods for the (1D)-fractional Laplacian. Mathematics of Computation, 87 (312). pp. 1821-1857. ISSN 0025-5718.

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This paper presents regularity results and associated high order numerical methods for one-dimensional fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight ω times a "regular" unknown, a characterization of the regularity of solutions is obtained in terms of the smoothness of the corresponding right-hand sides. In particular, for right-hand sides which are analytic in a Bernstein ellipse, analyticity in the same Bernstein ellipse is obtained for the ``regular'' unknown. Moreover, a sharp Sobolev regularity result is presented which completely characterizes the co-domain of the fractional-Laplacian operator in terms of certain weighted Sobolev spaces introduced in (Babuška and Guo, SIAM J. Numer. Anal. 2002). The present theoretical treatment relies on a full eigendecomposition for a certain weighted integral operator in terms of the Gegenbauer polynomial basis. The proposed Gegenbauer-based Nyström numerical method for the fractional-Laplacian Dirichlet problem, further, is significantly more accurate and efficient than other algorithms considered previously. The sharp error estimates presented in this paper indicate that the proposed algorithm is spectrally accurate, with convergence rates that only depend on the smoothness of the right-hand side. In particular, convergence is exponentially fast (resp. faster than any power of the mesh-size) for analytic (resp. infinitely smooth) right-hand sides. The properties of the algorithm are illustrated with a variety of numerical results.

Item Type:Article
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URLURL TypeDescription Paper
Bruno, Oscar0000-0001-8369-3014
Additional Information:© 2017 American Mathematical Society. Received by the editor August 30, 2016, and, in revised form, March 16, 2017. Article electronically published on November 9, 2017. This research was partially supported by CONICET under grant PIP 2014-2016 11220130100184CO. The work of the first author was partially supported by CONICET, Argentina, under grant PIP 2014–2016 11220130100184CO. The second and fourth author’s and MM’s efforts were made possible by a graduate fellowship from CONICET, Argentina. The third author’s efforts were supported by the US NSF and AFOSR through contracts DMS-1411876 and FA9550-15-1-0043, and by the NSSEFF Vannevar Bush Fellowship under contract number N00014-16-1-2808.
Funding AgencyGrant Number
Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)PIP 2014-2016 11220130100184CO
Air Force Office of Scientific Research (AFOSR)FA9550-15-1-0043
National Security Science and Engineering Faculty FellowshipN00014-16-1-2808
Subject Keywords:Fractional Laplacian, hypersingular integral equations, high order numerical methods, Gegenbauer polynomials
Issue or Number:312
Classification Code:2010 Mathematics Subject Classification. 65R20, 35B65, 33C45
Record Number:CaltechAUTHORS:20180502-133444210
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:86198
Deposited By: Tony Diaz
Deposited On:03 May 2018 21:05
Last Modified:03 Oct 2019 19:40

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