Computing Fractional Laplacians on Complex-Geometry Domains: Algorithms and Simulations

Song, Fangying; Xu, Chuanju; Karniadakis, George Em

doi:10.1137/16m1078197

Cited by 53 publications

(46 citation statements)

References 31 publications

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“…To discretize the Riesz fractional Laplacian, we use the adaptive finite element method (AFEM) of [71] and the Walk-on-Spheres (WOS) method of [22]. We discretize the spectral fractional Laplacian directly using the spectral element method (SEM) of [96], and the heat semigroup approach [36,19,37], which is used in Section 5, and elliptic extension approaches [33,37,36,80,81] are also discussed for completeness. We develop a new approximation method for the directional definition using a radial basis function (RBF) collocation method, which also makes use of the vector Grünwald scheme of [97].…”

Section: Section Overviewmentioning

confidence: 99%

“…We emphasize that these methods are all for the fractional Poisson problem with zero Dirichlet boundary conditions. For the computations in this article, we prefer to use the spectral element method of Song et al [96], described below, due to its accuracy and ease of implementation for the considered examples.…”

Section: Spectral Definitionmentioning

confidence: 99%

“…So this method is inexpensive in the context of time-dependent problems. For more details of the implementation and computational complexity for the SEM, see [96].…”

Section: Computational Considerationsmentioning

confidence: 99%

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Section: Spectral Definitionmentioning

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“…For such problems the spectral methods are known to be very accurate due to exponential convergence rate with respect to the number of the degrees of freedom. Such examples on square domains are presented in [36]. Alghough, the case of spectral approximation on a disk domain is discussed there, the application of their discretization to the fractional power problem would require computing the generalized eigenvectors for which fast methods are not available.…”

Section: Introductionmentioning

confidence: 99%

Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation

Harizanov

Lazarov

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et al. 2020

Journal of Computational Physics

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Here we study theoretically and compare experimentally with the methods developed in [19,8] an efficient method for solving systems of algebraic equations A α u h = f h , 0 < α < 1, where A is an N × N matrix coming from the discretization of a fractional diffusion operator. More specifically, we focus on matrices obtained from finite difference or finite element approximation of second order elliptic problems in R d , d = 1, 2, 3. The proposed methods are based on the best uniform rational approximation (BURA) r α,k (t) of t α on [0, 1]. Here r α,k is a rational function of t involving numerator and denominator polynomials of degree at most k.The approximation ofWe show that the proposed method is exponentially convergent with respect to k and has some attractive properties. First, it reduces the solution of the nonlocal system to solution of k systems with matrix ( A + cj I) and cj > 0, j = 1, 2, . . . , k. Thus, good computational complexity can be achieved if fast solvers are available for such systems. Second, the original problem and its rational approximation in the finite difference case are positivity preserving. In the finite element case, this valid for schemes obtained by mass lumping under certain mild conditions on the mesh. Further, we prove that the lumped mass schemes still have the expected rate of convergence, at times assuming additional regularity on the right hand side. Finally, we present comprehensive numerical experiments on a number of model problems for various α in one and two spatial dimensions. These illustrate the computational behavior of the proposed method and compare its accuracy and efficiency with that of other methods developed by Harizanov et. al. [19] and Bonito and Pasciak [8] .

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“…Other important numerical approaches found in the literature include the matrix transfer technique introduced by Ilić et al in [43,44] (which relies on computing an approximation to the fractional power of a matrix representing the discretization of the standard elliptic operator), the work by Song et al [61] (based on an efficient method to approximate the eigenpairs of the standard Laplacian on complex geometries), the articles by Vabishchevich [64,65] (where the solution of (1.1)-(1.2) is viewed and computed as the value of the solution to a suitable pseudo-parabolic problem at a specific point in time), and the works by Nochetto et al [52,23] (relying on the use of a Dirichlet-to-Neumann map to represent the fractional operator). Regarding these last works in fact, it is well-known [20,19,62] that for a given u, the solution v(x, y) to the extended problem…”

Section: Introductionmentioning

confidence: 99%

Numerical approximations for fractional elliptic equationsviathe method of semigroups

2020

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We provide a novel approach to the numerical solution of the family of nonlocal elliptic equations (−∆) s u = f in Ω, subject to some homogeneous boundary conditions B(u) = 0 on ∂Ω, where s ∈ (0, 1), Ω ⊂ R n is a bounded domain, and (−∆) s is the spectral fractional Laplacian associated to B on ∂Ω. We use the solution representation (−∆) −s f together with its singular integral expression given by the method of semigroups. By combining finite element discretizations for the heat semigroup with monotone quadratures for the singular integral we obtain accurate numerical solutions. Roughly speaking, given a datum f in a suitable fractional Sobolev space of order r ≥ 0 and the discretization parameter h > 0, our numerical scheme converges as O(h r+2s ), providing super quadratic convergence rates up to O(h 4 ) for sufficiently regular data, or simply O(h 2s ) for merely f ∈ L 2 (Ω). We also extend the proposed framework to the case of nonhomogeneous boundary conditions and support our results with some illustrative numerical tests.

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Computing Fractional Laplacians on Complex-Geometry Domains: Algorithms and Simulations

Cited by 53 publications

References 31 publications

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

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

Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation

Numerical approximations for fractional elliptic equationsviathe method of semigroups

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