Preconditioners for Fractional Diffusion Equations Based on the Spectral Symbol
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It is well known that the discretization of fractional diffusion equations (FDE) with fractional derivatives α∈(1,2), using the so-called weighted and shifted Grünwald formula, leads to linear systems whose coefficient matrices show a Toeplitz-like structure. More precisely, in the case of variable coefficients, the related matrix sequences belong to the so called Generalized Locally Toeplitz (GLT) class. Conversely, when the given FDE is with constant coefficients, using a suitable discretization, we encounter a Toeplitz structure associated to a nonnegative function F_α, called the spectral symbol, having a unique zero at zero of real positive order between one and two. For the fast solution of such systems by preconditioned Krylov methods, several preconditioning techniques have been proposed in both the one and two dimensional case. In this note we propose a new preconditioner denoted by P_F_α which belongs to the τ algebra and its based on the spectral symbol F_α. Comparing with some of the previously proposed preconditioners, we show that although the low band structure preserving preconditioners are more effective in the one dimensional case, the new preconditioner performs better in the more challenging two dimensional setting.
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