A Preconditioned Fast Collocation Method for a Linear Nonlocal Diffusion Model in Convex Domains

Recently, there are many papers dedicated to develop fast numerical methods for nonlocal diffusion and peridynamic models. However, these methods require the physical domain where we solve the governing equations is rectangular. To relax this restriction, in this article, we develop a fast collocation method for a static linear nonlocal diffusion model in convex domains via a volume-penalization approach. Based on the analysis of the structure of the coefficient matrix, we also present a fast solution technique of the linear system arising from the collocation discretization by accelerating the matrix-vector multiplications in the usual Krylov subspace iteration method. This technique helps to reduce the computational work in each Krylov subspace iteration from <inline-formula> <tex-math notation="LaTeX">$O(N^{2})$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$O(N\log N)$ </tex-math></inline-formula> and the storage requirement of the coefficient matrix from <inline-formula> <tex-math notation="LaTeX">$O(N^{2})$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$O(N)$ </tex-math></inline-formula> without any lossy compression, where <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> is the number of unknowns. Moreover, an efficient preconditioner was also provided in this work to accelerate the convergence of the Krylov subspace iteration method. Numerical experiments show the applicability of this method.

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