What is the fractional Laplacian? A comparative review with new results

The fractional Laplacian in Rd, which we write as (−Δ)α/2 with α∈(0,2), has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature...

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Veröffentlicht in:Journal of computational physics Jg. 404; S. 109009
Hauptverfasser: Lischke, Anna, Pang, Guofei, Gulian, Mamikon, Song, Fangying, Glusa, Christian, Zheng, Xiaoning, Mao, Zhiping, Cai, Wei, Meerschaert, Mark M., Ainsworth, Mark, Karniadakis, George Em
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cambridge Elsevier Inc 01.03.2020
Elsevier Science Ltd
Elsevier
Schlagworte:
Anomalous diffusion
Approximation
Boundary conditions
Boundary value problems
Collocation methods
Computational physics
Domains
First principles
Fractional Laplacian
MATHEMATICS AND COMPUTING
Nonlocal model
Numerical analysis
Numerical methods
Operators (mathematics)
Poisson equation
Radial basis function
Regularity
Spectra
Stable Lévy motion
Stable Lévy motion
Nonlocal model
Fractional Laplacian
Regularity
Anomalous diffusion
ISSN:0021-9991, 1090-2716
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Zusammenfassung:The fractional Laplacian in Rd, which we write as (−Δ)α/2 with α∈(0,2), has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. In contrast, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: “What is the fractional Laplacian?” Beginning from first principles, we compare several commonly used definitions of the fractional Laplacian theoretically, through their stochastic interpretations as well as their analytical properties. Then, we present quantitative comparisons using a sample of state-of-the-art methods. We discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems: radial basis function collocation (for the Riesz fractional Laplacian) and nonharmonic lifting (for the spectral fractional Laplacian). In our numerical studies, we aim to compare different definitions on bounded domains using a collection of benchmark problems. We consider the fractional Poisson equation with both zero and nonzero boundary conditions, where the fractional Laplacian is defined according to the Riesz definition, the spectral definition, the directional definition, and the horizon-based nonlocal definition. We verify the accuracy of the numerical methods used in the approximations for each operator, and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions. Through our efforts, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications. •Thorough theoretical discussion from first principles to current research topics.•Identification of different features in solutions to fractional Poisson problems.•Equivalence of operators and implementation for nonzero Dirichlet conditions.•A new RBF collocation method for the directional fractional Laplacian.•Discussion of stochastic connections for the various fractional Laplacians.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
AC04-94AL85000; NA0003525
SAND-2019-13611J
National Science Foundation (NSF)
USDOE National Nuclear Security Administration (NNSA)
National Natural Science Foundation of China (NNSFC)
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2019.109009