Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations
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...
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| Format: | Journal article |
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Society for Industrial and Applied Mathematics
2018
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| _version_ | 1797056586217160704 |
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| author | McDonald, E Pestana, J Wathen, A |
| author_facet | McDonald, E Pestana, J Wathen, A |
| author_sort | McDonald, E |
| collection | OXFORD |
| description | 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 |
| first_indexed | 2024-03-06T19:24:40Z |
| format | Journal article |
| id | oxford-uuid:1b4c6ea1-bb34-446a-8a88-8cd02a807c51 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T19:24:40Z |
| publishDate | 2018 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:1b4c6ea1-bb34-446a-8a88-8cd02a807c512022-03-26T10:59:32ZPreconditioning and iterative solution of all-at-once systems for evolutionary partial differential equationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:1b4c6ea1-bb34-446a-8a88-8cd02a807c51Symplectic Elements at OxfordSociety for Industrial and Applied Mathematics2018McDonald, EPestana, JWathen, AStandard 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 |
| spellingShingle | McDonald, E Pestana, J Wathen, A Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations |
| title | Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations |
| title_full | Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations |
| title_fullStr | Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations |
| title_full_unstemmed | Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations |
| title_short | Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations |
| title_sort | preconditioning and iterative solution of all at once systems for evolutionary partial differential equations |
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