CSC   24412
CENTRO DE SIMULACION COMPUTACIONAL PARA APLICACIONES TECNOLOGICAS
Unidad Ejecutora - UE
artículos
Título:
Regularity theory and high order numerical methods for the (1D)-fractional Laplacian
Autor/es:
J. P. BORTHAGARAY; G. ACOSTA; O. BRUNO; M. MAAS
Revista:
MATHEMATICS OF COMPUTATION
Editorial:
AMER MATHEMATICAL SOC
Referencias:
Lugar: Providence; Año: 2018 vol. 87
ISSN:
0025-5718
Resumen:
This paper presents regularity results and associated high-ordernumerical methods for one-dimensional Fractional-Laplacian boundary-valueproblems. 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 codomain of the Fractional-Laplacian operatorin terms of certain weighted Sobolev spaces introduced in (Babuska and Guo, SIAM J. Numer. Anal. 2002). The present theoretical treatment relieson a full eigen decomposition for a certain weighted integral operator in terms ofthe Gegenbauer polynomial basis. The proposed Gegenbauer-based Nystrom 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.