Multilevel algorithms considered as iterative methods on indefinite systems:
Abstract: "For the representation of piecewise d-linear functions we introduce a generating system instead of the usual finite element basis. It contains the nodal basis functions of the finest level of discretization and, additionally, the nodal basis functions of all coarser levels of discret...
Saved in:
| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
München
1991
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| Series: | Technische Universität <München>: TUM-I
9143 |
| Subjects: | |
| Summary: | Abstract: "For the representation of piecewise d-linear functions we introduce a generating system instead of the usual finite element basis. It contains the nodal basis functions of the finest level of discretization and, additionally, the nodal basis functions of all coarser levels of discretization. This approach enables us to work directly with multilevel decompositions of a function. For a partial differential equation, the Galerkin approach results now in an indefinite system of linear equations which has in the 1D case only about twice, in the 2D case about 4/3 times and in the 3D case about 8/7 times as many unknowns as the usual system. Furthermore, the indefinite system does not possess just one but many solutions However, the unique solution of the usual definite finite element problem can be computed from every solution of the indefinite problem. We show that modern mutlilevel algorithms can be considered as standard iterative methods over the indefinite system. The conjugate gradient method for the indefinite system is equivalent to the BPX- or MDS- preconditioned conjugate gradient method for the linear system which arises from the usual finite element basis. The Gauss-Seidel iteration applied to the indefinite system is equivalent to the multigrid method applied to the standard basis system. Consequently, the Gauss-Seidel- preconditioned conjugate gradient method for the indefinite system is equivalent to MG-CG for the standard basis system At last, the results of numerical experiments regarding the condition number and the convergence rates of different iterative methods for the indefinite system are reported. |
| Physical Description: | 31 S. graph. Darst. |
Staff View
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| 245 | 1 | 0 | |a Multilevel algorithms considered as iterative methods on indefinite systems |c Michael Griebel |
| 264 | 1 | |a München |c 1991 | |
| 300 | |a 31 S. |b graph. Darst. | ||
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| 490 | 1 | |a Technische Universität <München>: TUM-I |v 9143 | |
| 490 | 1 | |a Sonderforschungsbereich Methoden und Werkzeuge für die Nutzung paralleler Rechnerarchitekturen: SFB-Bericht / A |v 1991,29 | |
| 520 | 3 | |a Abstract: "For the representation of piecewise d-linear functions we introduce a generating system instead of the usual finite element basis. It contains the nodal basis functions of the finest level of discretization and, additionally, the nodal basis functions of all coarser levels of discretization. This approach enables us to work directly with multilevel decompositions of a function. For a partial differential equation, the Galerkin approach results now in an indefinite system of linear equations which has in the 1D case only about twice, in the 2D case about 4/3 times and in the 3D case about 8/7 times as many unknowns as the usual system. Furthermore, the indefinite system does not possess just one but many solutions | |
| 520 | 3 | |a However, the unique solution of the usual definite finite element problem can be computed from every solution of the indefinite problem. We show that modern mutlilevel algorithms can be considered as standard iterative methods over the indefinite system. The conjugate gradient method for the indefinite system is equivalent to the BPX- or MDS- preconditioned conjugate gradient method for the linear system which arises from the usual finite element basis. The Gauss-Seidel iteration applied to the indefinite system is equivalent to the multigrid method applied to the standard basis system. Consequently, the Gauss-Seidel- preconditioned conjugate gradient method for the indefinite system is equivalent to MG-CG for the standard basis system | |
| 520 | 3 | |a At last, the results of numerical experiments regarding the condition number and the convergence rates of different iterative methods for the indefinite system are reported. | |
| 650 | 4 | |a Differential equations, Partial | |
| 650 | 4 | |a Galerkin methods | |
| 810 | 2 | |a A |t Sonderforschungsbereich Methoden und Werkzeuge für die Nutzung paralleler Rechnerarchitekturen: SFB-Bericht |v 1991,29 |w (DE-604)BV004627888 |9 1991,29 | |
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| id | DE-604.BV005593601 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:32:03Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003503635 |
| oclc_num | 27742848 |
| open_access_boolean | |
| owner | DE-12 DE-91G DE-BY-TUM |
| owner_facet | DE-12 DE-91G DE-BY-TUM |
| physical | 31 S. graph. Darst. |
| publishDate | 1991 |
| publishDateSearch | 1991 |
| publishDateSort | 1991 |
| record_format | marc |
| series | Technische Universität <München>: TUM-I |
| series2 | Technische Universität <München>: TUM-I Sonderforschungsbereich Methoden und Werkzeuge für die Nutzung paralleler Rechnerarchitekturen: SFB-Bericht / A |
| spelling | Griebel, Michael 1960- Verfasser (DE-588)111660068 aut Multilevel algorithms considered as iterative methods on indefinite systems Michael Griebel München 1991 31 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Technische Universität <München>: TUM-I 9143 Sonderforschungsbereich Methoden und Werkzeuge für die Nutzung paralleler Rechnerarchitekturen: SFB-Bericht / A 1991,29 Abstract: "For the representation of piecewise d-linear functions we introduce a generating system instead of the usual finite element basis. It contains the nodal basis functions of the finest level of discretization and, additionally, the nodal basis functions of all coarser levels of discretization. This approach enables us to work directly with multilevel decompositions of a function. For a partial differential equation, the Galerkin approach results now in an indefinite system of linear equations which has in the 1D case only about twice, in the 2D case about 4/3 times and in the 3D case about 8/7 times as many unknowns as the usual system. Furthermore, the indefinite system does not possess just one but many solutions However, the unique solution of the usual definite finite element problem can be computed from every solution of the indefinite problem. We show that modern mutlilevel algorithms can be considered as standard iterative methods over the indefinite system. The conjugate gradient method for the indefinite system is equivalent to the BPX- or MDS- preconditioned conjugate gradient method for the linear system which arises from the usual finite element basis. The Gauss-Seidel iteration applied to the indefinite system is equivalent to the multigrid method applied to the standard basis system. Consequently, the Gauss-Seidel- preconditioned conjugate gradient method for the indefinite system is equivalent to MG-CG for the standard basis system At last, the results of numerical experiments regarding the condition number and the convergence rates of different iterative methods for the indefinite system are reported. Differential equations, Partial Galerkin methods A Sonderforschungsbereich Methoden und Werkzeuge für die Nutzung paralleler Rechnerarchitekturen: SFB-Bericht 1991,29 (DE-604)BV004627888 1991,29 Technische Universität <München>: TUM-I 9143 (DE-604)BV006185376 9143 |
| spellingShingle | Griebel, Michael 1960- Multilevel algorithms considered as iterative methods on indefinite systems Technische Universität <München>: TUM-I Differential equations, Partial Galerkin methods |
| title | Multilevel algorithms considered as iterative methods on indefinite systems |
| title_auth | Multilevel algorithms considered as iterative methods on indefinite systems |
| title_exact_search | Multilevel algorithms considered as iterative methods on indefinite systems |
| title_full | Multilevel algorithms considered as iterative methods on indefinite systems Michael Griebel |
| title_fullStr | Multilevel algorithms considered as iterative methods on indefinite systems Michael Griebel |
| title_full_unstemmed | Multilevel algorithms considered as iterative methods on indefinite systems Michael Griebel |
| title_short | Multilevel algorithms considered as iterative methods on indefinite systems |
| title_sort | multilevel algorithms considered as iterative methods on indefinite systems |
| topic | Differential equations, Partial Galerkin methods |
| topic_facet | Differential equations, Partial Galerkin methods |
| volume_link | (DE-604)BV004627888 (DE-604)BV006185376 |
| work_keys_str_mv | AT griebelmichael multilevelalgorithmsconsideredasiterativemethodsonindefinitesystems |