@TechReport{ it:2015-002,
author = {Marco Donatelli and Mariarosa Mazza and Stefano
Serra-Capizzano},
title = {Spectral Analysis and Structure Preserving Preconditioners
for Fractional Diffusion Equations},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2015},
number = {2015-002},
month = jan,
abstract = {Fractional partial order diffusion equations are a
generalization of classical partial differential equations,
used to model anomalous diffusion phenomena. When using the
implicit Euler formula and the shifted Gr{\"u}nwald
formula, it has been shown that the related discretizations
lead to a linear system whose coefficient matrix has a
Toeplitz-like structure. In this paper we focus our
attention on the case of variable diffusion coefficients.
Under appropriate conditions, we show that the sequence of
the coefficient matrices belongs to the Generalized Locally
Toeplitz class and we compute the symbol describing its
asymptotic eigenvalue distribution, as the matrix size
diverges. We employ the spectral information for analyzing
known methods of preconditioned Krylov and multigrid type,
with both positive and negative results and with a look
forward to the multidimensional setting. We also propose
two new tridiagonal structure preserving preconditioners to
solve the resulting linear system, with Krylov methods such
as CGNR and GMRES. A number of numerical examples shows
that our proposal is more effective than recently used
circulant preconditioner.}
}