A sine transform-based preconditioner for the fourth-order scheme arising from multi-dimensional nonlocal Poisson equations

Abstract

In this paper, a simple and easy-to-implement fourth-order fractional central difference (4FCD) method is used to discretize the multi-dimensional nonlocal Poisson equation involving the integral fractional Laplacian (IFL), which gives a multilevel symmetric and positive definite Toeplitz linear system. To efficiently solve the system, we propose a sine transform-based preconditioner and prove that the preconditioned conjugate gradient (PCG) method can achieve a convergence rate independent of mesh-size. Finally, numerical results are presented to demonstrate the effectiveness of the proposed preconditioner compared with state-of-the-art methods.