A fast collocation method for a variable-coefficient nonlocal diffusion model

We develop a fast collocation scheme for a variable-coefficient nonlocal diffusion model, for which a numerical discretization would yield a dense stiffness matrix. The development of the fast method is achieved by carefully handling the variable coefficients appearing inside the singular integral operator and exploiting the structure of the dense stiffness matrix. The resulting fast method reduces the computational work from O(N3) required by a commonly used direct solver to O(Nlog⁡N) per iteration and the memory requirement from O(N2) to O(N). Furthermore, the fast method reduces the computational work of assembling the stiffness matrix from O(N2) to O(N). Numerical results are presented to show the utility of the fast method.