Author
McDonald, E
Pestana, J
Wathen, A
Journal title
SIAM Journal on Scientific Computing
DOI
10.1137/16M1062016
Issue
2
Volume
40
Last updated
2024-03-26T12:03:12.83+00:00
Page
A1012-A1033
Abstract
Standard Krylov subspace solvers for self-adjoint problems have rigorous convergence bounds based solely on eigenvalues. However, for non-self-adjoint problems, eigenvalues do not determine behavior even for widely used iterative methods. In this paper, we discuss time-dependent PDE problems, which are always non-self-adjoint. We propose a block circulant preconditioner for the all-at-once evolutionary PDE system which has block Toeplitz structure. Through reordering of variables to obtain a symmetric system, we are able to rigorously establish convergence bounds for MINRES which guarantee a number of iterations independent of the number of time-steps for the all-at-once system. If the spatial differential operators are simultaneously diagonalizable, we are able to quickly apply the preconditioner through use of a sine transform; and for those that are not, we are able to use an algebraic multigrid process to provide a good approximation. Results are presented for solution to both the heat and convection diffusion equations. Read More: https://epubs.siam.org/doi/abs/10.1137/16M1062016
Symplectic ID
822219
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Publication type
Journal Article
Publication date
03 Apr 2018
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