An accurate and asymptotically compatible collocation scheme for nonlocal diffusion problems

作者：Zhang, Xiaoping; Wu, Jiming; Ju, Lili^*

来源：Applied Numerical Mathematics, 2018, 133: 52-68.

DOI：10.1016/j.apnum.2017.11.007

• Nonlocal diffusion problem
• Kernel function
• Quadrature rule
• Collocation scheme
• Asymptotic compatibility

In this paper, we develop and analyze a collocation scheme for solving the linear nonlocal diffusion problem with general kernels. To approximate the nonlocal diffusion operator, we take a classic trapezoidal rule based on the linear interpolation as the starting point, and then carefully derive a new improved quadrature rule, which is not only more accurate but also could avoid the evaluations of singular integrals. We then use this rule to construct a collocation scheme for solving the nonlocal diffusion equations, that produces a symmetric positive definite stiffness matrix with Toeplitz structure. The proposed scheme is rigorously shown to be of second order accurate with respect to the mesh size for the nonlocal problem with fixed horizon, and in particular, it can achieve higher order accuracy for the commonly used kernels in the literature. Furthermore, we also prove that the scheme is asymptotically compatible, i.e., the approximate solution of the nonlocal diffusion problem converges to the exact solution of the corresponding local PDE problem when the horizon and the mesh size both go to zero. Finally, numerical experiments are presented to verify the theoretical results.

• 出版日期2018-11
• 单位武汉大学; 北京计算科学研究中心; 北京应用物理与计算数学研究所