Sharp error estimates of a spectral Galerkin method for a diffusion-reaction equation with integral fractional Laplacian on a disk

Hao, Zhaopeng; Li, Huiyuan; Zhang, Zhimin; Zhang, Zhongqiang

doi:10.1090/mcom/3645

Sharp error estimates of a spectral Galerkin method for a diffusion-reaction equation with integral fractional Laplacian on a disk
HTML articles powered by AMS MathViewer

by Zhaopeng Hao, Huiyuan Li, Zhimin Zhang and Zhongqiang Zhang HTML | PDF

Math. Comp. 90 (2021), 2107-2135 Request permission

Abstract:

We investigate a spectral Galerkin method for the two-dimensional fractional diffusion-reaction equations on a disk. We first prove regularity estimates of solutions in the weighted Sobolev space. Then we obtain optimal convergence orders of a spectral Galerkin method for the fractional diffusion-reaction equations in the $L^2$ and energy norm. We present numerical results to verify the theoretical analysis.

References

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
Gabriel Acosta, Juan Pablo Borthagaray, Oscar Bruno, and Martín Maas, Regularity theory and high order numerical methods for the (1D)-fractional Laplacian, Math. Comp. 87 (2018), no. 312, 1821–1857. MR 3787393, DOI 10.1090/mcom/3276
Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Mark Ainsworth and Christian Glusa, Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains, Contemporary computational mathematics—a celebration of the 80th birthday of Ian Sloan. Vol. 1, 2, Springer, Cham, 2018, pp. 17–57. MR 3822227
George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145, DOI 10.1137/1.9781611970470
Andrea Bonito, Wenyu Lei, and Joseph E. Pasciak, Numerical approximation of the integral fractional Laplacian, Numer. Math. 142 (2019), no. 2, 235–278. MR 3941931, DOI 10.1007/s00211-019-01025-x
John P. Boyd and Fu Yu, Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan-Shepp ridge polynomials, Chebyshev-Fourier series, cylindrical Robert functions, Bessel-Fourier expansions, square-to-disk conformal mapping and radial basis functions, J. Comput. Phys. 230 (2011), no. 4, 1408–1438. MR 2753370, DOI 10.1016/j.jcp.2010.11.011
Domingos M. Cardoso, Paula Carvalho, Maria Aguieiras A. de Freitas, and Cybele T. M. Vinagre, Spectra, signless Laplacian and Laplacian spectra of complementary prisms of graphs, Linear Algebra Appl. 544 (2018), 325–338. MR 3765790, DOI 10.1016/j.laa.2018.01.020
Peter Constantin and Jiahong Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal. 30 (1999), no. 5, 937–948. MR 1709781, DOI 10.1137/S0036141098337333
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
Siwei Duo, Hans Werner van Wyk, and Yanzhi Zhang, A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem, J. Comput. Phys. 355 (2018), 233–252. MR 3738575, DOI 10.1016/j.jcp.2017.11.011
Bartłomiej Dyda, Alexey Kuznetsov, and Mateusz Kwaśnicki, Eigenvalues of the fractional Laplace operator in the unit ball, J. Lond. Math. Soc. (2) 95 (2017), no. 2, 500–518. MR 3656279, DOI 10.1112/jlms.12024
B. P. Epps and B. Cushman-Roisin, Turbulence modeling via the fractional Laplacian, 2018, http://www.dartmouth.edu/~cushman/papers/2018-JFM-FractionalLaplacian.pdf.
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
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
Zhaopeng Hao, Guang Lin, and Zhongqiang Zhang, Error estimates of a spectral Petrov-Galerkin method for two-sided fractional reaction-diffusion equations, Appl. Math. Comput. 374 (2020), 125045, 13. MR 4060532, DOI 10.1016/j.amc.2020.125045
Zhaopeng Hao and Zhongqiang Zhang, Optimal regularity and error estimates of a spectral Galerkin method for fractional advection-diffusion-reaction equations, SIAM J. Numer. Anal. 58 (2020), no. 1, 211–233. MR 4049399, DOI 10.1137/18M1234679
Zhaopeng Hao, Zhongqiang Zhang, and Rui Du, Fractional centered difference scheme for high-dimensional integral fractional Laplacian, J. Comput. Phys. 424 (2021), Paper No. 109851, 17. MR 4157658, DOI 10.1016/j.jcp.2020.109851
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
Mateusz Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal. 20 (2017), no. 1, 7–51. MR 3613319, DOI 10.1515/fca-2017-0002
H. Li, Direct Galerkin spectral methods using orthogonal polynomials on the disc, 2019. In preparation.
Huiyuan Li and Yuan Xu, Spectral approximation on the unit ball, SIAM J. Numer. Anal. 52 (2014), no. 6, 2647–2675. MR 3276427, DOI 10.1137/130940591
Anna Lischke, Guofei Pang, Mamikon Gulian et al., What is the fractional Laplacian? A comparative review with new results, J. Comput. Phys. 404 (2020), 109009, 62. MR 4043885, DOI 10.1016/j.jcp.2019.109009
Michael Loss and Craig Sloane, Hardy inequalities for fractional integrals on general domains, J. Funct. Anal. 259 (2010), no. 6, 1369–1379. MR 2659764, DOI 10.1016/j.jfa.2010.05.001
Victor Minden and Lexing Ying, A simple solver for the fractional Laplacian in multiple dimensions, SIAM J. Sci. Comput. 42 (2020), no. 2, A878–A900. MR 4079475, DOI 10.1137/18M1170406
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
Gabor Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, New York, 1939. MR 0000077, DOI 10.1090/coll/023
Xiaochuan Tian, Qiang Du, and Max Gunzburger, Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains, Adv. Comput. Math. 42 (2016), no. 6, 1363–1380. MR 3571209, DOI 10.1007/s10444-016-9466-z
Alex Townsend, Marcus Webb, and Sheehan Olver, Fast polynomial transforms based on Toeplitz and Hankel matrices, Math. Comp. 87 (2018), no. 312, 1913–1934. MR 3787396, DOI 10.1090/mcom/3277
Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503903
W. T. M. Verkley, A spectral model for two-dimensional incompressible fluid flow in a circular basin. I. Mathematical formulation, J. Comput. Phys. 136 (1997), no. 1, 100–114. MR 1468626, DOI 10.1006/jcph.1997.5747
Hong Wang and Kaixin Wang, An $O(N\log ^2N)$ alternating-direction finite difference method for two-dimensional fractional diffusion equations, J. Comput. Phys. 230 (2011), no. 21, 7830–7839. MR 2825722, DOI 10.1016/j.jcp.2011.07.003
Alfred Wünsche, Generalized Zernike or disc polynomials, J. Comput. Appl. Math. 174 (2005), no. 1, 135–163. MR 2102653, DOI 10.1016/j.cam.2004.04.004
Zhongqiang Zhang, Error estimates of spectral Galerkin methods for a linear fractional reaction-diffusion equation, J. Sci. Comput. 78 (2019), no. 2, 1087–1110. MR 3918681, DOI 10.1007/s10915-018-0800-0