An Efficient Collocation Method for a Non-local Diffusion Model

The non-local diffusion model provides an appropriate description of the deformation of a continuous body involving discontinuities or other singularities, which cannot be described properly by classical theory of solid mechanics. However, because the non-local nature of the non-local diffusion operator, the numerical methods for non-local diffusion model generate dense or even full stiffness matrices. A direct solver typically requires O(N) of operations and O(N) of memory where N is the number of unknowns. We develop a fast collocation method for the non-local diffusion model which has the following features: (i) It reduces the computational cost from O(N) to O(N log N) and memory requirement from O(N) to O(N). (ii) It requires only one-fold integration in the evaluation of the stiffness matrix. Numerical experiments show the utility of the method.