The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator

Authors:
Marta D'elia;Max Gunzburger
Affiliations:
-;-
Venue:
Computers & Mathematics with Applications
Year:
2013

Citing 13
Cited 0

An efficient finite element method for treating singularities in Laplace's equation

Journal of Computational Physics
Finite difference approximations for fractional advection-dispersion flow equations

Journal of Computational and Applied Mathematics
Weighted average finite difference methods for fractional diffusion equations

Journal of Computational Physics
Limits of dense graph sequences

Journal of Combinatorial Theory Series B
Vectorized adaptive quadrature in MATLAB

Journal of Computational and Applied Mathematics
Numerical treatment of fractional heat equations

Applied Numerical Mathematics
Matrix approach to discrete fractional calculus II: Partial fractional differential equations

Journal of Computational Physics
Numerical inversion of 2-D Laplace transforms applied to fractional diffusion equations

Applied Numerical Mathematics
Least squares finite-element solution of a fractional order two-point boundary value problem

Computers & Mathematics with Applications
Image Recovery via Nonlocal Operators

Journal of Scientific Computing
On Learning with Integral Operators

The Journal of Machine Learning Research
Image Denoising Methods. A New Nonlocal Principle

SIAM Review
Analysis of an Asymptotic Preserving Scheme for Linear Kinetic Equations in the Diffusion Limit

SIAM Journal on Numerical Analysis

Quantified Score

Hi-index	0.09

Visualization

Abstract

We analyze a nonlocal diffusion operator having as special cases the fractional Laplacian and fractional differential operators that arise in several applications. In our analysis, a nonlocal vector calculus is exploited to define a weak formulation of the nonlocal problem. We demonstrate that, when sufficient conditions on certain kernel functions hold, the solution of the nonlocal equation converges to the solution of the fractional Laplacian equation on bounded domains as the nonlocal interactions become infinite. We also introduce a continuous Galerkin finite element discretization of the nonlocal weak formulation and we derive a priori error estimates. Through several numerical examples we illustrate the theoretical results and we show that by solving the nonlocal problem it is possible to obtain accurate approximations of the solutions of fractional differential equations circumventing the problem of treating infinite-volume constraints.