  In the finite difference approximation of the fractional Laplacian the
stiffness matrix is typically dense and needs to be approximated numerically.
The effect of the accuracy in approximating the stiffness matrix on the
accuracy in the whole computation is analyzed and shown to be significant. Four
such approximations are discussed. While they are shown to work well with the
recently developed grid-over finite difference method (GoFD) for the numerical
solution of boundary value problems of the fractional Laplacian, they differ in
accuracy, economics to compute, performance of preconditioning, and asymptotic
decay away from the diagonal line. In addition, two preconditioners based on
sparse and circulant matrices are discussed for the iterative solution of
linear systems associated with the stiffness matrix. Numerical results in two
and three dimensions are presented.
