An Efficient and Accurate Numerical Method for the Spectral Fractional Laplacian Equation

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 $$d+1$$ 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.

[1]  Jie Shen,et al.  Enriched Spectral Methods and Applications to Problems with Weakly Singular Solutions , 2018, Journal of Scientific Computing.

[2]  David M. Miller,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[3]  K. Miller,et al.  Completely monotonic functions , 2001 .

[4]  ShenJie.  Efficient spectral-Galerkin method I , 1994 .

[5]  Mark Ainsworth,et al.  Hybrid Finite Element-Spectral Method for the Fractional Laplacian: Approximation Theory and Efficient Solver , 2018, SIAM J. Sci. Comput..

[6]  Dale B. Haidvogel,et al.  The Accurate Solution of Poisson's Equation by Expansion in Chebyshev Polynomials , 1979 .

[7]  Jie Shen,et al.  Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials , 1994, SIAM J. Sci. Comput..

[8]  Ricardo H. Nochetto,et al.  Numerical methods for fractional diffusion , 2017, Comput. Vis. Sci..

[9]  John R. Rice,et al.  Direct solution of partial difference equations by tensor product methods , 1964 .

[10]  P. R. Stinga,et al.  Extension Problem and Harnack's Inequality for Some Fractional Operators , 2009, 0910.2569.

[11]  L. Caffarelli,et al.  An Extension Problem Related to the Fractional Laplacian , 2006, math/0608640.

[12]  Yannick Sire,et al.  Nonlinear equations for fractional laplacians II: existence, uniqueness, and qualitative properties of solutions , 2011, 1111.0796.

[13]  Louis Dupaigne,et al.  Regularity of Radial Extremal Solutions for Some Non-Local Semilinear Equations , 2010, 1004.1906.

[14]  Jie Shen,et al.  Spectral Methods: Algorithms, Analysis and Applications , 2011 .

[15]  Jie Shen,et al.  Generalized Jacobi polynomials/functions and their applications , 2009 .

[16]  M. Landsberg,et al.  K. Yosida, Functional Analysis. (Die Grundlehren der mathematischen Wissenschaften, Band 123) XII + 458 S. Berlin/Heidelberg/New York 1965. Springer‐Verlag. Preis geb. DM 66, — , 1966 .

[17]  G. Ben-yu,et al.  Generalized Laguerre Interpolation and Pseudospectral Method for Unbounded Domains , 2006 .

[18]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[19]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[20]  Jie Shen,et al.  Generalized Jacobi functions and their applications to fractional differential equations , 2014, Math. Comput..

[21]  Harbir Antil,et al.  Model reduction for fractional elliptic problems using Kato's formula , 2019 .

[22]  Ricardo H. Nochetto,et al.  A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis , 2013, Found. Comput. Math..

[23]  C. Brändle,et al.  A concave—convex elliptic problem involving the fractional Laplacian , 2010, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[24]  Ben-yu Guo,et al.  Generalized Laguerre Interpolation and Pseudospectral Method for Unbounded Domains , 2006, SIAM J. Numer. Anal..

[25]  Jie Shen,et al.  Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials , 2006, J. Sci. Comput..

[26]  E. Valdinoci,et al.  Hitchhiker's guide to the fractional Sobolev spaces , 2011, 1104.4345.

[27]  Ricardo H. Nochetto,et al.  Multilevel methods for nonuniformly elliptic operators and fractional diffusion , 2016, Math. Comput..

[28]  Ricardo H. Nochetto,et al.  A PDE Approach to Space-Time Fractional Parabolic Problems , 2014, SIAM J. Numer. Anal..

[29]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .