Abstract
We propose in this paper an efficient and accurate numerical method for the spectral fractional Laplacian equation using the Caffarelli–Silvestre extension. In particular, we propose several strategies to deal with the singularity and the additional dimension associated with the extension problem: (i) reducing the \(d+1\) dimensional problem to a sequence of d-dimension Poisson-type problems by using the matrix diagonalizational method; (ii) resolving the singularity by applying the enriched spectral method in the extended dimension. We carry out rigorous analysis for the proposed numerical method, and provide abundant numerical examples to verify the theoretical results and illustrate effectiveness of the proposed method.
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References
Abramowitz, M., Stegun, I.A., Miller, D.: Handbook of Mathematical Functions With Formulas. Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55). US GPO (1965)
Ainsworth, M., Glusa, C.: Hybrid finite element-spectral method for the fractional Laplacian: approximation theory and efficient solver. SIAM J. Sci. Comput. 40(4), A2383–A2405 (2018)
Bonito, A., Borthagaray, J.P., Nochetto, R.H., Otárola, E., Salgado, A.J.: Numerical methods for fractional diffusion. Comput. Vis. Sci. 19, 1–28 (2018)
Brändle, C., Colorado, E., de Pablo, A., Sánchez, U.: A concave convex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinb. 143(01), 39–71 (2013)
Cabre, X., Sire, Y.: Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. Trans. Am. Math. Soc. 367(2), 911–941 (2014)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Part. Differ. Equ. 32(8), 1245–1260 (2007)
Capella, A., Dávila, J., Dupaigne, L., Sire, Y.: Regularity of radial extremal solutions for some non-local semilinear equations. Commun. Part. Differ. Equ. 36(8), 1353–1384 (2011)
Chen, L., Nochetto, R.H., Otárola, E., Salgado, A.J.: Multilevel methods for nonuniformly elliptic operators and fractional diffusion. Math. Comput. 85(302), 1 (2016)
Chen, S., Shen, J.: Enriched spectral methods and applications to problems with weakly singular solutions. J. Sci. Comput. 77(3), 1468–1489 (2018)
Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603–1638 (2016)
Dinh, H., Antil, H., Chen, Y., Cherkaev, E., Narayan, A.: Model reduction for fractional elliptic problems using Kato’s formula. arXiv:1904.09332 (2019)
Evans, L.: Partial Differential Equations, vol. 19, 2nd edn. American Mathematical Society, Providence (2010)
Guo, B.Y., Shen, J., Wang, L.L.: Optimal spectral-galerkin methods using generalized Jacobi polynomials. J. Sci. Comput. 27(1), 305–322 (2006)
Guo, B.Y., Shen, J., Wang, L.L.: Generalized Jacobi polynomials/functions and their applications. Appl. Numer. Math. 59(5), 1011–1028 (2009)
Guo, B.Y., Wang, L.L., Wang, Z.Q.: Generalized Laguerre interpolation and pseudospectral method for unbounded domains. SIAM J. Numer. Anal. 43(6), 2567–2589 (2006)
Haidvogel, D.B., Zang, T.: The accurate solution of Poisson’s equation by expansion in Chebyshev polynomials. J. Comput. Phys. 30(2), 167–180 (1979)
Lunardi, A.: Interpolation Theory, 2nd edn. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). Edizioni della Normale, Pisa (2009)
Lynch, R.E., Rice, J.R., Thomas, D.H.: Direct solution of partial difference equations by tensor product methods. Numer. Math. 6(1), 185–199 (1964)
Miller, K.S., Samko, S.G.: Completely monotonic functions. Integr. Transf. Spec. Funct. 12(4), 389–402 (2001)
Nezza, E.D., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Nochetto, R.H., Otárola, E., Salgado, A.J.: A PDE approach to space-time fractional parabolic problems. SIAM J. Numer. Anal. 54(2), 848–873 (2016)
Nochetto, R.H., Otárola, E., Salgado, A.J.: A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15(3), 733–791 (2014)
Shen, J.: Efficient spectral-galerkin method i. direct solvers of second-and fourth-order equations using legendre polynomials. SIAM J. Sci. Comput. 15(6), 1489–1505 (1994)
Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)
Stinga, P.R., Torrea, J.L.: Extension problem and Harnack’s inequality for some fractional operators. Commun. Part. Differ. Equ. 35(11), 2092–2122 (2010)
Szego, G.: Orthogonal Polynomials, vol. 23 of amer. In Math. Soc. Colloq. Publ., Amer. Math. Soc., Providence (1975)
Yosida, K.: Functional Analysis. Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123. Springer, New York (1968)
Acknowledgements
J.S. would like to thank Professor Ricardo Nochetto for stimulating discussions.
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S.C. is partially supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. BK20181002), the National Natural Science Foundation for the Youth of China (Grant No. 11801235) and the Postdoctoral Science Foundation of China (Grant Nos. BX20180032, 2019M650459).
J.S. is partially supported by NSFC 11971407 and 91630204, and NSF DMS-1720442.
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Chen, S., Shen, J. An Efficient and Accurate Numerical Method for the Spectral Fractional Laplacian Equation. J Sci Comput 82, 17 (2020). https://doi.org/10.1007/s10915-019-01122-x
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DOI: https://doi.org/10.1007/s10915-019-01122-x