Barakitis, Nikos; Ekström, Sven-Erik; Vassalos, Paris Preconditioners for fractional diffusion equations based on the spectral symbol. (English) Zbl 1549.65251 Numer. Linear Algebra Appl. 29, No. 5, e2441, 22 p. (2022). Given is a fractional diffusion equation for \(u(x,y,t)\) on a square \((x,y)\in\Omega\) \[ u'=\mathcal{D}u+f,~~\mathcal{D}=d_+D_{+x}^{\alpha}+d_-D_{-x}^{\alpha}+e_+D_{+y}^{\beta}+e_-D_{-y}^{\beta}, \] for \(t\in(0,T)\) with boundary and initial conditions. The \(\pm\) refer to left or right Liouville-type space-fractional derivatives of order \(\alpha,\beta\in(1,2)\).First it is explained how the first- and second-order discretization lead to a linear system \(Mu=b\) to be solved at each time step where \(M\) involves Toeplitz matrices (their size depends on \(x\)- and \(y\)-discretizations) with a simple symbol.The main results are low-band preserving preconditioners for \(M\) that are based on the symbols of symmetric positive definite Toeplitz matrices, both in the 1-D and 2-D case. See [D. Noutsos et al., Linear Algebra Appl. 491, 276–291 (2016; Zbl 1391.65071)].The preconditioned spectrum is proved to be bounded under some restrictive conditions on \(d_\pm\) and \(e_\pm\). Numerical examples show that the proposed preconditioners are slightly less effective in 1-D but they outperform known ones in 2-D. Reviewer: Adhemar Bultheel (Leuven) Cited in 6 Documents MSC: 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35R11 Fractional partial differential equations 65F08 Preconditioners for iterative methods 65F10 Iterative numerical methods for linear systems 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 65F35 Numerical computation of matrix norms, conditioning, scaling 26A33 Fractional derivatives and integrals 15B05 Toeplitz, Cauchy, and related matrices Keywords:Liouville fractional derivative; preconditioning; fractional differential equations; multi-level Toeplitz matrix; fractional order zero; GMRES; sine transform based preconditioner Citations:Zbl 1391.65071 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] MeerschaertMM, TadjeranC. 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