Linear and Nonlinear Fractional Elliptic Problems

Borthagaray, Juan Pablo; Li, Wenbo; Nochetto, Ricardo H.

doi:10.48550/arxiv.1906.04230

Cited by 3 publications

(3 citation statements)

References 53 publications

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“…We mention multigrid preconditioners, [AG17] based on uniformly refined mesh hierarchies and operator preconditioning, [Hip06,GSUT19,SvV19], which requires one to realize an operator of the opposite order. Another, classical technique is the framework of additive Schwarz preconditioners, analyzed in a BPX-setting with Fourier techniques in [BLN19]. In the present work, we also adopt the additive Schwarz framework and show that, also in the presence of adaptively refined meshes, multilevel diagonal scaling leads to uniformly bounded condition numbers for the integral fractional Laplacian.…”

Section: Introductionmentioning

confidence: 79%

On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

Faustmann¹,

Melenk²,

Parvizi³

2019

Preprint

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We provide an endpoint stability result for Scott-Zhang type operators in Besov spaces. For globally continuous piecewise polynomials these are bounded from H 3/2 into B 3/2 2,∞ ; for elementwise polynomials these are bounded from H 1/2 into B 1/2 2,∞ . As an application, we obtain a multilevel decomposition based on Scott-Zhang operators on a hierarchy of meshes generated by newest vertex bisection with equivalent norms up to (but excluding) the endpoint case. A local multilevel diagonal preconditioner for the fractional Laplacian on locally refined meshes with optimal eigenvalue bounds is presented.

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Section: Introductionmentioning

confidence: 79%

On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

Faustmann¹,

Melenk²,

Parvizi³

2019

Preprint

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“…For smooth domains Ω ⊂ R d with ∂Ω ∈ C ∞ , the shift theorem in Assumption 2.1 holds for any ε > 0, see, e.g., [Gru15]. For polygonal Lipschitz domains, which are of interest in applications, a similar shift theorem is mentioned in [BLN19] as part of the forthcoming work [BN].…”

Section: The Model Problemmentioning

confidence: 99%

Local convergence of the FEM for the integral fractional Laplacian

Faustmann¹,

Karkulik²,

Melenk³

2020

Preprint

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“…A non-conforming discretization, based on a Dunford-Taylor representation was proposed and analyzed in [6]. We refer to [5,8] for further discussion on these methods. In contrast, the analysis of finite difference schemes typically leads to error estimates in the L ∞ (Ω)-norm under regularity assumptions that cannot be guaranteed in general [16,17,26].…”

Section: Introductionmentioning

confidence: 99%

Local energy estimates for the fractional Laplacian

Borthagaray¹,

Leykekhman²,

Nochetto³

2020

Preprint

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The integral fractional Laplacian of order s ∈ (0, 1) is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result the global convergence rate of the numerical solutions. For finite element discretizations, we derive local error estimates in the H s -seminorm and show optimal convergence rates in the interior of the domain by only assuming meshes to be shape-regular. These estimates quantify the fact that the reduced approximation error is concentrated near the boundary of the domain. We illustrate our theoretical results with several numerical examples.

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Linear and Nonlinear Fractional Elliptic Problems

Cited by 3 publications

References 53 publications

On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

Local convergence of the FEM for the integral fractional Laplacian

Local energy estimates for the fractional Laplacian

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