TY - JOUR
T1 - A Mixed-Type Circulant Preconditioner for a Nonlocal Elastic Model
AU - Huang , Yuan-Yuan
AU - Lei , Siu-Long
JO - East Asian Journal on Applied Mathematics
VL - 4
SP - 791
EP - 812
PY - 2023
DA - 2023/10
SN - 13
DO - http://doi.org/10.4208/eajam.2021-275.210722
UR - https://global-sci.org/intro/article_detail/eajam/22063.html
KW - Generalized nonlocal elastic model, peridynamic, fractional differential operator, Toeplitz linear system, circulant preconditioner.
AB - <p style="text-align: justify;">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.</p>