TY - JOUR
T1 - Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations
AU - Siu-Long Lei, Xu Chen & Xinhe Zhang
JO - East Asian Journal on Applied Mathematics
VL - 2
SP - 109
EP - 130
PY - 2018
DA - 2018/02
SN - 6
DO - http://doi.org/10.4208/eajam.060815.180116a
UR - https://global-sci.org/intro/article_detail/eajam/10777.html
KW - High-dimensional two-sided fractional diffusion equation, implicit finite difference method, unconditionally stable, multilevel circulant preconditioner, GMRES method.
AB - <p style="text-align: justify;">High-dimensional two-sided space fractional diffusion equations with variable
diffusion coefficients are discussed. The problems can be solved by an implicit
finite difference scheme that is proven to be uniquely solvable, unconditionally stable
and first-order convergent in the infinity norm. A nonsingular multilevel circulant preconditoner
is proposed to accelerate the convergence rate of the Krylov subspace linear
system solver efficiently. The preconditoned matrix for fast convergence is a sum of the
identity matrix, a matrix with small norm, and a matrix with low rank under certain
conditions. Moreover, the preconditioner is practical, with an O(N logN) operation cost
and O(N) memory requirement. Illustrative numerical examples are also presented.</p>