Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian

Faustmann, Markus; Melenk, Jens Markus; Praetorius, Dirk

doi:10.1090/mcom/3603

Cited by 17 publications

(14 citation statements)

References 46 publications

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“…Whereas it may be possible to construct an appropriately graded mesh for simple geometries, the general case will require adaptively refined meshes to resolve the boundary singularity. A posteriori error estimators for the integral fractional Laplacian have been developed in Nochetto, von Petersdorff and Zhang (2010), Faustmann, Melenk, Parvizi and Praetorius (2019) and Ainsworth and Glusa (2017).…”

Section: Finite Element Methods For the Integral Fractional Laplacianmentioning

confidence: 99%

Numerical methods for nonlocal and fractional models

et al. 2020

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Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

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Section: Finite Element Methods For the Integral Fractional Laplacianmentioning

confidence: 99%

Numerical methods for nonlocal and fractional models

et al. 2020

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“…The replacement of the Bank-Weiser estimator by an anisotropic a posteriori error estimator would improve the convergence rate even further in case of boundary layers, see e.g. [17,57], Another interesting extension would be to test our method on fractional powers of other kinds of elliptic operators, following [23], on another definition of the fractional Laplacian operator [29] and/or other boundary conditions, following [14].…”

Section: Discussionmentioning

confidence: 99%

“…in [3,66,77]. Among the methods addressing the above numerical issues, we can cite: methods to efficiently solve eigenvalue problems [40], multigrid methods for performing efficient dense matrixvector products [6,7], hybrid finite element-spectral schemes [8], Dirichlet-to-Neumann maps (such as the Caffarelli-Silvestre extension) [13,36,44,57,75,79], semigroups methods [46,45,78], rational approximation methods [1,65,67], Dunford-Taylor integrals [21,26,25,28,23,31,60,67] (which can be considered as particular examples of rational approximation methods) and reduced basis methods [47,48,51].…”

Section: Introductionmentioning

confidence: 99%

“…Unlike the standard Laplacian equation, solutions to the fractional Laplacian problems often exhibit strong boundary layers even for smooth data, particularly when the fractional power is low [62]. These singularities lead to computational difficulties and have to be taken into account using, for example a priori geometric mesh refinement towards the boundary of the domain [4,18,29,31,57], or partition of unity enrichments [30]. We emphasize that [23] contains already an a priori error analysis in the L 2 norm for the combined rational sum finite element method that we use in this work.…”

Section: Introductionmentioning

confidence: 99%

“…In [27,44] the authors present another estimator, based on the solution to local problems on cylindrical stars, for the integral fractional Laplacian discretized using the Caffarelli-Silvestre extension. A weighted residual estimator is derived in [58] in the same context.…”

Section: Introductionmentioning

confidence: 99%

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An a posteriori error estimator for the spectral fractional power of the Laplacian

Bulle,

Barrera,

Bordas

et al. 2022

Preprint

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We develop a novel a posteriori error estimator for the L 2 error committed by the finite element discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes advantage of the semi-discretization scheme using a rational approximation which allows to reformulate the fractional problem into a family of non-fractional parametric problems. The estimator involves applying the implicit Bank-Weiser error estimation strategy to each parametric non-fractional problem and reconstructing the fractional error through the same rational approximation used to compute the solution to the original fractional problem. We provide several numerical examples in both two and three-dimensions demonstrating the effectivity of our estimator for varying fractional powers and its ability to drive an adaptive mesh refinement strategy.

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