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

Authors: Gabriel Acosta, Juan Pablo Borthagaray, Oscar Bruno and Martín Maas
Journal: Math. Comp.
MSC (2010): Primary 65R20, 35B65, 33C45
Published electronically: November 9, 2017
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Abstract: 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 $ \omega $ 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.

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Additional Information

Gabriel Acosta
Affiliation: IMAS - CONICET and Departamento de Matemática, FCEyN - Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I (1428) Buenos Aires, Argentina

Juan Pablo Borthagaray
Affiliation: IMAS - CONICET and Departamento de Matemática, FCEyN - Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I (1428) Buenos Aires, Argentina

Oscar Bruno
Affiliation: California Institute of Technology, Pasadena, California

Martín Maas
Affiliation: IAFE - CONICET and Departamento de Matemática, FCEyN - Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I (1428) Buenos Aires, Argentina

Keywords: Fractional Laplacian, hypersingular integral equations, high order numerical methods, Gegenbauer polynomials
Received by editor(s): August 30, 2016
Received by editor(s) in revised form: March 16, 2017
Published electronically: November 9, 2017
Additional Notes: 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.
Article copyright: © Copyright 2017 American Mathematical Society

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