Kronecker product-based structure preserving preconditioner for three-dimensional space-fractional diffusion equations

ABSTRACT

This paper is concerned with the construction of efficient preconditioner for systems arising from discretization of a three-dimensional space-fractional diffusion equations. The matrix structure of the resulting linear systems is the summation of several 3-level Kronecker products. We propose to use one 3-level Kronecker product as a preconditioner. We obtain this Kronecker product through an alternating splitting iteration method, which is shown to be convergent. Optimal values of the iteration parameters are also obtained. The splitting iteration is then accelerated by a Krylov subspace method like GMRES. The components of the 3-level Kronecker product preconditioner have the same structure as the matrix derived from discretization of one-dimensional problem. Therefore, we use structure preserving approximation to the discrete one-dimensional problem as the building block for our preconditioner. Several numerical experiments are presented to show the effectiveness of our approach.

KEYWORDS:

• Space fractional diffusion equation
• finite difference method
• preconditioning
• Kronecker product splitting

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65F10
• 65L06
• 65N22

Acknowledgments

The authors are very grateful to the referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11301575], the Chongqing Municipal Education Commission [grant numbers KJ1703055, KJZD-M201800501 and KJQN201800506], the Natural Science Foundation of Chongqing [grant numbers cstc2018jcyjAX0113 and cstc2018jcyjAX0794], and the Talent Project of Chongqing Normal University [grant number 02030307-0054].