A Fast Preconditioned Algorithm for Nonlocal Diffusion Model

Abstract:

A fast collocation scheme can be used to discretize the variable-coef?cient nonlocal diffusion model effectively. The coefficient matrix of the resulting linear system is unsymmetrical， dense and Toeplitz-like. The generalized minimum residual （GMRES） method can be employed to solve the discretized linear systems. In order to improve the rate of convergence of the GMRES method， the Toeplitz preconditioner and circulant preconditioner are constructed for the coefficient matrix， and the preconditioned GMRES methods are proposed for solving the discretized linear systems. Numerical examples are presented to illustrate the effectiveness of the preconditioned methods.

Key words:nonlocal diffusion model;collocation method;Toeplitz matrix;preconditioning;generalized minimum residual method