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

Acosta, Gabriel; Borthagaray, Juan Pablo; Bruno, Oscar; Maas, Martín

doi:10.1090/mcom/3276

Regularity theory and high order numerical methods for the (1D)-fractional Laplacian
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by Gabriel Acosta, Juan Pablo Borthagaray, Oscar Bruno and Martín Maas PDF

Math. Comp. 87 (2018), 1821-1857 Request permission

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.

References

Nicola Abatangelo, Large $S$-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 5555–5607. MR 3393247, DOI 10.3934/dcds.2015.35.5555
Milton Abramowitz (ed.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1965. Superintendent of Documents. MR 0177136
Gabriel Acosta and Juan Pablo Borthagaray, A fractional Laplace equation: regularity of solutions and finite element approximations, SIAM J. Numer. Anal. 55 (2017), no. 2, 472–495. MR 3620141, DOI 10.1137/15M1033952
Guglielmo Albanese, Alessio Fiscella, and Enrico Valdinoci, Gevrey regularity for integro-differential operators, J. Math. Anal. Appl. 428 (2015), no. 2, 1225–1238. MR 3334976, DOI 10.1016/j.jmaa.2015.04.002
Ivo Babuška and Benqi Guo, Direct and inverse approximation theorems for the $p$-version of the finite element method in the framework of weighted Besov spaces. I. Approximability of functions in the weighted Besov spaces, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1512–1538. MR 1885705, DOI 10.1137/S0036142901356551
W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964. MR 0185155
D. A. Benson, S. W. Wheatcraft, and Mark M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), no. 6, 1403–1412.
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
C. Brändle, E. Colorado, A. de Pablo, and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), no. 1, 39–71. MR 3023003, DOI 10.1017/S0308210511000175
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
O. P. Bruno and S. K. Lintner, Second-kind integral solvers for TE and TM problems of diffraction by open arcs, Radio Science, DOI 10.1029/2012rs005035.
Luis Caffarelli and Luis Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245–1260. MR 2354493, DOI 10.1080/03605300600987306
P. Carr, H. Geman, D. B. Madan, and M. Yor, The fine structure of asset returns: An empirical investigation, The Journal of Business 75(2002), no. 2, 305–332.
Rama Cont and Peter Tankov, Financial modelling with jump processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2042661
Matteo Cozzi, Interior regularity of solutions of non-local equations in Sobolev and Nikol’skii spaces, Ann. Mat. Pura Appl. (4) 196 (2017), no. 2, 555–578. MR 3624965, DOI 10.1007/s10231-016-0586-3
Marta D’Elia and Max Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput. Math. Appl. 66 (2013), no. 7, 1245–1260. MR 3096457, DOI 10.1016/j.camwa.2013.07.022
Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. MR 2944369, DOI 10.1016/j.bulsci.2011.12.004
Bartłomiej Dyda, Alexey Kuznetsov, and Mateusz Kwaśnicki, Fractional Laplace operator and Meijer G-function, Constr. Approx. 45 (2017), no. 3, 427–448. MR 3640641, DOI 10.1007/s00365-016-9336-4
Paolo Gatto and Jan S. Hesthaven, Numerical approximation of the fractional Laplacian via $hp$-finite elements, with an application to image denoising, J. Sci. Comput. 65 (2015), no. 1, 249–270. MR 3394445, DOI 10.1007/s10915-014-9959-1
Guy Gilboa and Stanley Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul. 7 (2008), no. 3, 1005–1028. MR 2480109, DOI 10.1137/070698592
Gerd Grubb, Fractional Laplacians on domains, a development of Hörmander’s theory of $\mu$-transmission pseudodifferential operators, Adv. Math. 268 (2015), 478–528. MR 3276603, DOI 10.1016/j.aim.2014.09.018
Ben-yu Guo and Li-lian Wang, Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory 128 (2004), no. 1, 1–41. MR 2063010, DOI 10.1016/j.jat.2004.03.008
Nicholas Hale and Alex Townsend, Fast and accurate computation of Gauss-Legendre and Gauss-Jacobi quadrature nodes and weights, SIAM J. Sci. Comput. 35 (2013), no. 2, A652–A674. MR 3033086, DOI 10.1137/120889873
Yanghong Huang and Adam Oberman, Numerical methods for the fractional Laplacian: a finite difference–quadrature approach, SIAM J. Numer. Anal. 52 (2014), no. 6, 3056–3084. MR 3504596, DOI 10.1137/140954040
J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1937591
Joseph Klafter and Igor M. Sokolov, Anomalous diffusion spreads its wings, Physics world 18 (2005), no. 8, pages 29.
Rainer Kress, Linear integral equations, 3rd ed., Applied Mathematical Sciences, vol. 82, Springer, New York, 2014. MR 3184286, DOI 10.1007/978-1-4614-9593-2
T. A. M. Langlands, B. I. Henry, and S. L. Wearne, Fractional cable equation models for anomalous electrodiffusion in nerve cells: finite domain solutions, SIAM J. Appl. Math. 71 (2011), no. 4, 1168–1203. MR 2823498, DOI 10.1137/090775920
Stéphane K. Lintner and Oscar P. Bruno, A generalized Calderón formula for open-arc diffraction problems: theoretical considerations, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), no. 2, 331–364. MR 3327958, DOI 10.1017/S0308210512000807
Ralf Metzler and Joseph Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004), no. 31, R161–R208. MR 2090004, DOI 10.1088/0305-4470/37/31/R01
Ricardo H. Nochetto, Enrique Otárola, and Abner J. Salgado, A PDE approach to fractional diffusion in general domains: a priori error analysis, Found. Comput. Math. 15 (2015), no. 3, 733–791. MR 3348172, DOI 10.1007/s10208-014-9208-x
Xavier Ros-Oton and Joaquim Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), no. 3, 275–302 (English, with English and French summaries). MR 3168912, DOI 10.1016/j.matpur.2013.06.003
Walter Rudin, Principles of mathematical analysis, 2nd ed., McGraw-Hill Book Co., New York, 1964. MR 0166310
Youcef Saad and Martin H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986), no. 3, 856–869. MR 848568, DOI 10.1137/0907058
Raffaella Servadei and Enrico Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 4, 831–855. MR 3233760, DOI 10.1017/S0308210512001783
G. T. Symm, Integral equation methods in potential theory. II, Proc. Roy. Soc. London Ser. A 275 (1963), 33–46. MR 154076, DOI 10.1098/rspa.1963.0153
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
Enrico Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA 49 (2009), 33–44. MR 2584076
Ziqing Xie, Li-Lian Wang, and Xiaodan Zhao, On exponential convergence of Gegenbauer interpolation and spectral differentiation, Math. Comp. 82 (2013), no. 282, 1017–1036. MR 3008847, DOI 10.1090/S0025-5718-2012-02645-7
Y. Yan and I. H. Sloan, On integral equations of the first kind with logarithmic kernels, J. Integral Equations Appl. 1 (1988), no. 4, 549–579. MR 1008406, DOI 10.1216/JIE-1988-1-4-549
Xiaodan Zhao, Li-Lian Wang, and Ziqing Xie, Sharp error bounds for Jacobi expansions and Gegenbauer-Gauss quadrature of analytic functions, SIAM J. Numer. Anal. 51 (2013), no. 3, 1443–1469. MR 3053576, DOI 10.1137/12089421X