A Mixed-Type Circulant Preconditioner for a Nonlocal Elastic Model

Abstract

A linear system from finite difference discretization of a generalized nonlocal elastic model was studied, where the model is composed of a Riesz potential operator with a fractional differential operator. Some properties of the coefficient matrix are proven theoretically and it is found that the linear system is very ill-conditioned when the parameter in the long-range hydrodynamic interactions is close to zero. Therefore, the usual Krylov subspace method with the Strang-Strang circulant preconditioner loses the power of preconditioning so that the iterative method converges slowly. Here the problem is fixed by utilizing a mixed-type circulant preconditioner obtained by both Strang’s and Chan’s circulant approximations. The invertibility of the preconditioner and a small-norm-low-rank decomposition of the difference matrix of the coefficient matrix and the preconditioner are shown theoretically under certain conditions. Numerical examples are given to illustrate the efficiency of the proposed fast solver.