An Efficient Jacobi Spectral Collocation Method with Nonlocal Quadrature Rules for Multi-Dimensional Volume-Constrained Nonlocal Models

In this paper, an efficient Jacobi spectral collocation method is developed for multi-dimensional weakly singular volume-constrained nonlocal models including both nonlocal diffusion (ND) models and peridynamic (PD) models. The model equation contains a weakly singular integral operator with the singularity located at the center of the integral domain, and the numerical approximation of it becomes an essential difficulty in solving nonlocal models. To approximate such singular nonlocal integrals in an accurate way, a novel nonlocal quadrature rule is constructed to accurately compute these integrals for the numerical solutions produced by spectral methods. Numerical experiments are given to show that spectral accuracy can be obtained by using the proposed Jacobi spectral collocation methods for several different nonlocal models. Besides, we numerically verify that the numerical solution of our Jacobi spectral method can converge to its correct local limit as the nonlocal interactions vanish.