A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations

DOI10.1137/15M1033952zbMath1359.65246arXiv1507.08970OpenAlexW3123275897MaRDI QIDQ2968170

Gabriel Acosta, Juan Pablo Borthagaray

Publication date: 10 March 2017

Published in: SIAM Journal on Numerical Analysis (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1507.08970

zbMATH Keywords

convergencefinite elementsa priori estimatesnumerical examplesgraded meshesfractional Laplacianweighted fractional norms

Mathematics Subject Classification ID

Smoothness and regularity of solutions to PDEs (35B65) Error bounds for boundary value problems involving PDEs (65N15) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation (35J05) Fractional partial differential equations (35R11)

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Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator
• On fractional Poincaré inequalities
• Hitchhiker's guide to the fractional Sobolev spaces
• Fractional Laplacians on domains, a development of Hörmander's theory of \(\mu\)-transmission pseudodifferential operators
• Image recovery via nonlocal operators
• Numerical approximation of the fractional Laplacian via \(hp\)-finite elements, with an application to image denoising
• Censored stable processes
• Finite element approximation for the fractional eigenvalue problem
• A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian
• A fractional order Hardy inequality
• Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. II: The three-dimensional case
• Reformulation of elasticity theory for discontinuities and long-range forces
• A PDE approach to fractional diffusion in general domains: a priori error analysis
• The Dirichlet problem for the fractional Laplacian: regularity up to the boundary
• Analysis of the Scott–Zhang interpolation in the fractional order Sobolev spaces
• On the spectrum of two different fractional operators
• From the long jump random walk to the fractional Laplacian
• On Fractional Laplacians
• Extension Problem and Harnack's Inequality for Some Fractional Operators
• Nonlocal Operators with Applications to Image Processing
• First Passage Times for Symmetric Stable Processes in Space
• Finite Element Interpolation of Nonsmooth Functions Satisfying Boundary Conditions
• Definitions for uniform domains
• Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary elements methods. Part I. The two-dimensional case
• The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics
• Financial Modelling with Jump Processes
• A NONLOCAL VECTOR CALCULUS, NONLOCAL VOLUME-CONSTRAINED PROBLEMS, AND NONLOCAL BALANCE LAWS
• The integrability of negative powers of the soution to the Saint Venant problem
• Image Denoising Methods. A New Nonlocal Principle
• Numerical Methods for the Fractional Laplacian: A Finite Difference-Quadrature Approach
• An Extension Problem Related to the Fractional Laplacian
• Variational formulation for the stationary fractional advection dispersion equation
• From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion
• Boundary Element Methods

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