  We develop a fast divided-and-conquer indirect collocation method for the
homogeneous Dirichlet boundary value problem of variable-order space-fractional
diffusion equations. Due to the impact of the space-dependent variable order,
the resulting stiffness matrix of the numerical approximation does not have a
Toeplitz-like structure. In this paper we derive a fast approximation of the
coefficient matrix by the means of a sum of Toeplitz matrices multiplied by
diagonal matrices. We show that the approximation is asymptotically consistent
with the original problem, which requires $O(kN\log^2 N)$ memory and $O(k
N\log^3 N)$ computational complexity with $N$ and $k$ being the numbers of
unknowns and the approximants, respectively. Numerical experiments are
presented to demonstrate the effectiveness and the efficiency of the proposed
method.
