Asymptotic Singular Value Analysis of the BSC Preconditioning for Solving Space Fractional Diffusion Equations

Abstract

Based on the Toeplitz structure contained in the banded preconditioner with shift compensation (BSC preconditioner), the asymptotic singular value distribution of the BSC-preconditioned matrix for solving the space fractional diffusion equations with nonequal diffusion coefficients is analyzed by exploiting the theory of the generalized locally Toeplitz (GLT) sequence. The theoretical analysis illustrates that the conditioning of the BSC-preconditioned matrix is bounded by

\mathcal{O}(h^{-\beta (2-\beta )/2})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal O(h^{-\beta (2-\beta )/2})$$\end{document}

when

\beta

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta $$\end{document}

is sufficiently close to 2, and is bounded by

\mathcal{O}(h^{\beta -1})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal O(h^{\beta -1})$$\end{document}

when

\beta

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta $$\end{document}

is sufficiently close to 1. Numerical computation also verifies that the BSC preconditioner is robust for

\beta

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta $$\end{document}

sufficiently approaching 1 or 2.

Keywords

Banded preconditioner with shift compensation (BSC preconditioner) / Generalized locally Toeplitz (GLT) / Asymptotic singular value distribution / 65F08 / 65F10 / 65M22 / 65N06 / 65N22 / 65Z05 / CR: G1.3

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Funding

National Natural Science Foundation of China(12001048)

RIGHTS & PERMISSIONS

Shanghai University