Fast collocation method for a two-dimensional variable-coefficient linear nonlocal diffusion model

Xuhao Zhang,Aijie Cheng

doi:10.1186/s13662-020-02931-3

Abstract

In this paper, a fast collocation method is developed for a two-dimensional variable-coefficient linear nonlocal diffusion model. By carefully dealing with the variable coefficient in the integral operator and then analyzing the structure of the coefficient matrix, we can reduce the computational operations in each Krylov subspace iteration from O(N^{2}) to O(Nlog N) and the memory requirement for the coefficient matrix from O(N^{2}) to O(N). Numerical experiments are carried out to show the utility of the fast collocation method.

Highlights

1 Introduction Recently, nonlocal models such as fractional partial differential equations (FPDEs) and nonlocal diffusion and peridynamic models have been applied in many research fields
The fast methods developed before do not apply to the variable-coefficient peridynamic model. To overcome this difficulty, [23] developed a fast collocation method to a one-dimensional variablecoefficient nonlocal diffusion model based on a piecewise-constant approximation to the variable coefficient
5 Conclusions In this paper, a fast and faithful collocation method is developed for a two-dimensional variable-coefficient linear nonlocal diffusion model

Summary

Introduction

Nonlocal models such as fractional partial differential equations (FPDEs) and nonlocal diffusion and peridynamic models have been applied in many research fields. There have been many papers aimed to develop fast numerical schemes for nonlocal models to reduce the computational complexity and the memory requirement [13, 15,16,17,18,19,20,21]. In [18], the authors developed a fast collocation method for a two-dimensional linear nonlocal diffusion model. The fast methods developed before do not apply to the variable-coefficient peridynamic model.

Results

Conclusion