An All-at-Once Preconditioner for Evolutionary Partial Differential Equations

Xue Lei Lin, Michael Ng

Research output: Contribution to journalJournal articlepeer-review

22 Citations (Scopus)

Abstract

In [McDonald, Pestana, and Wathen, SIAM J. Sci. Comput., 40 (2018), pp. A1012--A1033], a block circulant preconditioner is proposed for all-at-once linear systems arising from evolutionary partial differential equations, in which the preconditioned matrix is proven to be diagonalizable and to have identity-plus-low-rank decomposition in the case of the heat equation. In this paper, we generalize the block circulant preconditioner by introducing a small parameter 𝜖>0 into the top-right block of the block circulant preconditioner. The implementation of the generalized preconditioner requires the same computational complexity as that of the block circulant one. Theoretically, we prove that (i) the generalization preserves the diagonalizability and the identity-plus-low-rank decomposition; (ii) all eigenvalues of the new preconditioned matrix are clustered at 1 for sufficiently small 𝜖; (iii) GMRES method for the preconditioned system has a linear convergence rate independent of size of the linear system when 𝜖 is taken to be smaller than or comparable to square root of time-step size. Numerical results are reported to confirm the efficiency of the proposed preconditioner and to show that the generalization improves the performance of block circulant preconditioner.

Original languageEnglish
Pages (from-to)A2766-A2784
Number of pages19
JournalSIAM Journal on Scientific Computing
Volume43
Issue number4
DOIs
Publication statusPublished - Jan 2021

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • All-at-once discretization
  • Block Toeplitz matrices
  • Convergence of GMRES
  • Evolutionary equations
  • Preconditioning technique

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