TY - JOUR
T1 - An Efficient Collocation Method for a Non-Local Diffusion Model
AU - H. Tian, H. Wang & W. Wang
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 815
EP - 825
PY - 2013
DA - 2013/10
SN - 10
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/597.html
KW - collocation method, dense matrices, fast methods, non-local diffusion, peridynamics.
AB - <p style="text-align: justify;">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 $O(N^3)$ of operations and $O(N^2)$ 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 $O(N^3)$ to $O(N log^2 N)$ and memory requirement from $O(N^2)$ 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.</p>