Regularity theory and high order numerical methods for the (1D)-fractional Laplacian
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 o...
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todo:paper_00255718_v87_n312_p1821_Acosta2023-10-03T14:36:16Z Regularity theory and high order numerical methods for the (1D)-fractional Laplacian Acosta, G. Borthagaray, J.P. Bruno, O. Maas, M. Fractional Laplacian Gegenbauer polynomials High order numerical methods Hypersingular integral equations 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. © 2017 American Mathematical Society. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00255718_v87_n312_p1821_Acosta |
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Universidad de Buenos Aires |
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I-28 |
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R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Fractional Laplacian Gegenbauer polynomials High order numerical methods Hypersingular integral equations |
| spellingShingle |
Fractional Laplacian Gegenbauer polynomials High order numerical methods Hypersingular integral equations Acosta, G. Borthagaray, J.P. Bruno, O. Maas, M. Regularity theory and high order numerical methods for the (1D)-fractional Laplacian |
| topic_facet |
Fractional Laplacian Gegenbauer polynomials High order numerical methods Hypersingular integral equations |
| description |
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. © 2017 American Mathematical Society. |
| format |
JOUR |
| author |
Acosta, G. Borthagaray, J.P. Bruno, O. Maas, M. |
| author_facet |
Acosta, G. Borthagaray, J.P. Bruno, O. Maas, M. |
| author_sort |
Acosta, G. |
| title |
Regularity theory and high order numerical methods for the (1D)-fractional Laplacian |
| title_short |
Regularity theory and high order numerical methods for the (1D)-fractional Laplacian |
| title_full |
Regularity theory and high order numerical methods for the (1D)-fractional Laplacian |
| title_fullStr |
Regularity theory and high order numerical methods for the (1D)-fractional Laplacian |
| title_full_unstemmed |
Regularity theory and high order numerical methods for the (1D)-fractional Laplacian |
| title_sort |
regularity theory and high order numerical methods for the (1d)-fractional laplacian |
| url |
http://hdl.handle.net/20.500.12110/paper_00255718_v87_n312_p1821_Acosta |
| work_keys_str_mv |
AT acostag regularitytheoryandhighordernumericalmethodsforthe1dfractionallaplacian AT borthagarayjp regularitytheoryandhighordernumericalmethodsforthe1dfractionallaplacian AT brunoo regularitytheoryandhighordernumericalmethodsforthe1dfractionallaplacian AT maasm regularitytheoryandhighordernumericalmethodsforthe1dfractionallaplacian |
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1807321215365283840 |