Volume 4, Issue 4
Correlation Between Mesh Geometry and Stiffness Matrix Conditioning for Nonlocal Diffusion Models

Jibum Kim

East Asian J. Appl. Math., 4 (2014), pp. 312-328.

Published online: 2018-02

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  • Abstract

Nonlocal diffusion models involve integral equations that account for nonlocal interactions and do not explicitly employ differential operators in the space variables. Due to the nonlocality, they might look different from classical partial differential equation (PDE) models, but their local limit reduces to partial differential equations. The effect of mesh element anisotropy, mesh refinement and kernel functions on the conditioning of the stiffness matrix for a nonlocal diffusion model on 2D geometric domains is considered, and the results compared with those obtained from typical local PDE models. Numerical experiments show that the condition number is bounded by c−2 (where c is a constant) for an integrable kernel function, and is not affected by the choice of the basis function. In contrast to local PDE models, mesh anisotropy and refinement affect the condition number very little.

  • Keywords

65R20 65M60

  • AMS Subject Headings

Peridynamics condition number finite element method.

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-4-312, author = {Jibum Kim}, title = {Correlation Between Mesh Geometry and Stiffness Matrix Conditioning for Nonlocal Diffusion Models}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {4}, number = {4}, pages = {312--328}, abstract = {

Nonlocal diffusion models involve integral equations that account for nonlocal interactions and do not explicitly employ differential operators in the space variables. Due to the nonlocality, they might look different from classical partial differential equation (PDE) models, but their local limit reduces to partial differential equations. The effect of mesh element anisotropy, mesh refinement and kernel functions on the conditioning of the stiffness matrix for a nonlocal diffusion model on 2D geometric domains is considered, and the results compared with those obtained from typical local PDE models. Numerical experiments show that the condition number is bounded by c−2 (where c is a constant) for an integrable kernel function, and is not affected by the choice of the basis function. In contrast to local PDE models, mesh anisotropy and refinement affect the condition number very little.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.270114.280914a}, url = {http://global-sci.org/intro/article_detail/eajam/10840.html} }
TY - JOUR T1 - Correlation Between Mesh Geometry and Stiffness Matrix Conditioning for Nonlocal Diffusion Models AU - Jibum Kim JO - East Asian Journal on Applied Mathematics VL - 4 SP - 312 EP - 328 PY - 2018 DA - 2018/02 SN - 4 DO - http://doi.org/10.4208/eajam.270114.280914a UR - https://global-sci.org/intro/article_detail/eajam/10840.html KW - 65R20 KW - 65M60 AB -

Nonlocal diffusion models involve integral equations that account for nonlocal interactions and do not explicitly employ differential operators in the space variables. Due to the nonlocality, they might look different from classical partial differential equation (PDE) models, but their local limit reduces to partial differential equations. The effect of mesh element anisotropy, mesh refinement and kernel functions on the conditioning of the stiffness matrix for a nonlocal diffusion model on 2D geometric domains is considered, and the results compared with those obtained from typical local PDE models. Numerical experiments show that the condition number is bounded by c−2 (where c is a constant) for an integrable kernel function, and is not affected by the choice of the basis function. In contrast to local PDE models, mesh anisotropy and refinement affect the condition number very little.

Jibum Kim. (1970). Correlation Between Mesh Geometry and Stiffness Matrix Conditioning for Nonlocal Diffusion Models. East Asian Journal on Applied Mathematics. 4 (4). 312-328. doi:10.4208/eajam.270114.280914a
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