  A grid-overlay finite difference method is proposed for the numerical
approximation of the fractional Laplacian on arbitrary bounded domains. The
method uses an unstructured simplicial mesh and an overlay uniform grid for the
underlying domain and constructs the approximation based on a uniform-grid
finite difference approximation and a data transfer from the unstructured mesh
to the uniform grid. The method takes full advantages of both uniform-grid
finite difference approximation in efficient matrix-vector multiplication via
the fast Fourier transform and unstructured meshes for complex geometries and
mesh adaptation. It is shown that its stiffness matrix is similar to a
symmetric and positive definite matrix and thus invertible if the data transfer
has full column rank and positive column sums. Piecewise linear interpolation
is studied as a special example for the data transfer. It is proved that the
full column rank and positive column sums of linear interpolation is guaranteed
if the spacing of the uniform grid is smaller than or equal to a positive bound
proportional to the minimum element height of the unstructured mesh. Moreover,
a sparse preconditioner is proposed for the iterative solution of the resulting
linear system for the homogeneous Dirichlet problem of the fractional
Laplacian. Numerical examples demonstrate that the new method has similar
convergence behavior as existing finite difference and finite element methods
and that the sparse preconditioning is effective. Furthermore, the new method
can readily be incorporated with existing mesh adaptation strategies. Numerical
results obtained by combining with the so-called MMPDE moving mesh method are
also presented.
