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Volume 10, Issue 4
An Efficient Collocation Method for a Non-Local Diffusion Model

H. Tian, H. Wang & W. Wang

Int. J. Numer. Anal. Mod., 10 (2013), pp. 815-825.

Published online: 2013-10

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  • Abstract

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.

  • Keywords

collocation method, dense matrices, fast methods, non-local diffusion, peridynamics.

  • AMS Subject Headings

65J, 65R, 65N

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-10-815, author = {H. Tian, H. Wang and W. Wang}, title = {An Efficient Collocation Method for a Non-Local Diffusion Model}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {4}, pages = {815--825}, abstract = {

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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/597.html} }
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 -

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.

H. Tian, H. Wang and W. Wang. (2013). An Efficient Collocation Method for a Non-Local Diffusion Model. International Journal of Numerical Analysis and Modeling. 10 (4). 815-825. doi:
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