Preconditioning the Advection-Diffusion Equation: the Green's Function Approach
We look at the relationship between efficient preconditioners (i.e., good approximations to the discrete inverse operator) and the generalized inverse for the (continuous) advection-diffusion operator -- the Green's function. We find that the continuous Green's function exhibits two import...
| Main Authors: | , |
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| Format: | Report |
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Unspecified
1997
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| _version_ | 1797082909411115008 |
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| author | Loghin, D Wathen, A |
| author_facet | Loghin, D Wathen, A |
| author_sort | Loghin, D |
| collection | OXFORD |
| description | We look at the relationship between efficient preconditioners (i.e., good approximations to the discrete inverse operator) and the generalized inverse for the (continuous) advection-diffusion operator -- the Green's function. We find that the continuous Green's function exhibits two important properties -- directionality and rapid downwind decay -- which are preserved by the discrete (grid) Green's functions, if and only if the discretization used produces non-oscillatory solutions. In particular, the downwind decay ensures the locality of the grid Green's functions. Hence, a finite element formulation which produces a good solution will typically use a coefficient matrix with almost lower triangular structure under a "with-the-flow" numbering of the variables. It follows that the block Gauss-Seidel matrix is a first candidate for a preconditioner to use with an iterative solver of Krylov subspace type. |
| first_indexed | 2024-03-07T01:34:35Z |
| format | Report |
| id | oxford-uuid:94b480e7-0544-4735-8de8-23387688f4fe |
| institution | University of Oxford |
| last_indexed | 2024-03-07T01:34:35Z |
| publishDate | 1997 |
| publisher | Unspecified |
| record_format | dspace |
| spelling | oxford-uuid:94b480e7-0544-4735-8de8-23387688f4fe2022-03-26T23:41:18ZPreconditioning the Advection-Diffusion Equation: the Green's Function ApproachReporthttp://purl.org/coar/resource_type/c_93fcuuid:94b480e7-0544-4735-8de8-23387688f4feMathematical Institute - ePrintsUnspecified1997Loghin, DWathen, AWe look at the relationship between efficient preconditioners (i.e., good approximations to the discrete inverse operator) and the generalized inverse for the (continuous) advection-diffusion operator -- the Green's function. We find that the continuous Green's function exhibits two important properties -- directionality and rapid downwind decay -- which are preserved by the discrete (grid) Green's functions, if and only if the discretization used produces non-oscillatory solutions. In particular, the downwind decay ensures the locality of the grid Green's functions. Hence, a finite element formulation which produces a good solution will typically use a coefficient matrix with almost lower triangular structure under a "with-the-flow" numbering of the variables. It follows that the block Gauss-Seidel matrix is a first candidate for a preconditioner to use with an iterative solver of Krylov subspace type. |
| spellingShingle | Loghin, D Wathen, A Preconditioning the Advection-Diffusion Equation: the Green's Function Approach |
| title | Preconditioning the Advection-Diffusion Equation: the Green's Function Approach |
| title_full | Preconditioning the Advection-Diffusion Equation: the Green's Function Approach |
| title_fullStr | Preconditioning the Advection-Diffusion Equation: the Green's Function Approach |
| title_full_unstemmed | Preconditioning the Advection-Diffusion Equation: the Green's Function Approach |
| title_short | Preconditioning the Advection-Diffusion Equation: the Green's Function Approach |
| title_sort | preconditioning the advection diffusion equation the green s function approach |
| work_keys_str_mv | AT loghind preconditioningtheadvectiondiffusionequationthegreensfunctionapproach AT wathena preconditioningtheadvectiondiffusionequationthegreensfunctionapproach |