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Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications
In this paper, we propose accurate and efficient finite difference methods to
discretize the two- and three-dimensional fractional Laplacian
() in hypersingular integral
form. The proposed finite difference methods provide a fractional analogue of
the central difference schemes to the fractional Laplacian, and as , they collapse to the central difference schemes of the classical Laplace
operator . We prove that our methods are consistent if ,
and the local truncation error is , with a small constant and denoting the floor function. If
, they can achieve the
second order of accuracy for any . These results hold for
any dimension and thus improve the existing error estimates for the
finite difference method of the one-dimensional fractional Laplacian. Extensive
numerical experiments are provided and confirm our analytical results. We then
apply our method to solve the fractional Poisson problems and the fractional
Allen-Cahn equations. Numerical simulations suggest that to achieve the second
order of accuracy, the solution of the fractional Poisson problem should {\it
at most} satisfy . One merit of our methods is
that they yield a multilevel Toeplitz stiffness matrix, an appealing property
for the development of fast algorithms via the fast Fourier transform (FFT).
Our studies of the two- and three-dimensional fractional Allen-Cahn equations
demonstrate the efficiency of our methods in solving the high-dimensional
fractional problems.Comment: 24 pages, 6 figures, and 6 table
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