1,150 research outputs found
The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator
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.Comment: 27 pages, 5 figure
Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach
The fractional Laplacian is a non-local operator which
depends on the parameter and recovers the usual Laplacian as . A numerical method for the fractional Laplacian is proposed, based on
the singular integral representation for the operator. The method combines
finite difference with numerical quadrature, to obtain a discrete convolution
operator with positive weights. The accuracy of the method is shown to be
. Convergence of the method is proven. The treatment of far
field boundary conditions using an asymptotic approximation to the integral is
used to obtain an accurate method. Numerical experiments on known exact
solutions validate the predicted convergence rates. Computational examples
include exponentially and algebraically decaying solution with varying
regularity. The generalization to nonlinear equations involving the operator is
discussed: the obstacle problem for the fractional Laplacian is computed.Comment: 29 pages, 9 figure
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