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

TL;DR

This paper studies a nonlocal diffusion operator that includes the fractional Laplacian as a special case, providing a weak formulation, convergence analysis, finite element discretization, and numerical validation for approximating fractional differential equations.

Contribution

It introduces a unified framework for nonlocal operators, proves convergence to fractional Laplacian solutions, and develops a finite element method with error estimates for practical computation.

Findings

Nonlocal solutions converge to fractional Laplacian solutions under certain conditions.

Finite element discretization yields accurate approximations with proven error bounds.

Numerical examples confirm theoretical convergence and effectiveness of the approach.

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.