## Abstract

A fast accurate approximation method with multigrid solver is proposed to solve a two-dimensional fractional sub-diffusion equation. Using the finite difference discretization of fractional time derivative, a block lower triangular Toeplitz matrix is obtained where each main diagonal block contains a two-dimensional matrix for the Laplacian operator. Our idea is to make use of the block ϵ-circulant approximation via fast Fourier transforms, so that the resulting task is to solve a block diagonal system, where each diagonal block matrix is the sum of a complex scalar times the identity matrix and a Laplacian matrix. We show that the accuracy of the approximation scheme is of O(ϵ). Because of the special diagonal block structure, we employ the multigrid method to solve the resulting linear systems. The convergence of the multigrid method is studied. Numerical examples are presented to illustrate the accuracy of the proposed approximation scheme and the efficiency of the proposed solver.

Original language | English |
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Pages (from-to) | 204-218 |

Number of pages | 15 |

Journal | Journal of Computational Physics |

Volume | 323 |

DOIs | |

Publication status | Published - 15 Oct 2016 |

## Scopus Subject Areas

- Numerical Analysis
- Modelling and Simulation
- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

## User-Defined Keywords

- Block lower triangular Toeplitz matrix
- Block ϵ-circulant approximation
- Fractional sub-diffusion equations
- Multigrid method