An efficient multigrid method with preconditioned smoother for two-dimensional anisotropic space-fractional diffusion equations. (15th October 2022)

• Record Type: Journal Article
• Authors: Xu, Yuan; Lei, Siu-Long; Sun, Hai-Wei
• Is Part Of: Computers & mathematics with applications. Volume 124(2022)
• Page Start: 218
• Page End: 226
• Publication Date: 2022-10-15
• Subjects: Fractional diffusion equations -- Multigrid method -- Preconditioner -- Anisotropy; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2022.08.030
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: The anisotropic space-fractional diffusion equations in two dimensions are discretized by the Crank-Nicolson difference scheme with the weighted and shifted Grünwald formula, which is unconditionally stable and second-order convergence. The coefficient matrix of the discretized linear system possesses a two-level Toeplitz-like structure. Due to the anisotropy, the standard multigrid method converges slowly. By utilizing the GMRES method with a newly proposed tridiagonal preconditioner as a smoother, the convergence rate of the multigrid method can be accelerated significantly. The proposed tridiagonal preconditioner is shown to be invertible and a numerical experiment is given to demonstrate the efficiency of the proposed multigrid method with preconditioned smoother.