The Ritz Method for Boundary Problems with Essential Conditions as Constraints
We give an elementary derivation of an extension of the Ritz method to trial functions that do not satisfy essential boundary conditions. As in the Babuška-Brezzi approach boundary conditions are treated as variational constraints and Lagrange multipliers are used to remove them. However, we avoid t...
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2016/7058017 |
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| _version_ | 1850106563313795072 |
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| author | Vojin Jovanovic Sergiy Koshkin |
| author_facet | Vojin Jovanovic Sergiy Koshkin |
| author_sort | Vojin Jovanovic |
| collection | DOAJ |
| description | We give an elementary derivation of an extension of the Ritz method to trial functions that do not satisfy essential boundary conditions. As in the Babuška-Brezzi approach boundary conditions are treated as variational constraints and Lagrange multipliers are used to remove them. However, we avoid the saddle point reformulation of the problem and therefore do not have to deal with the Babuška-Brezzi inf-sup condition. In higher dimensions boundary weights are used to approximate the boundary conditions, and the assumptions in our convergence proof are stated in terms of completeness of the trial functions and of the boundary weights. These assumptions are much more straightforward to verify than the Babuška-Brezzi condition. We also discuss limitations of the method and implementation issues that follow from our analysis and examine a number of examples, both analytic and numerical. |
| format | Article |
| id | doaj-art-4e612bddf6b44eec854bd398ee72c657 |
| institution | OA Journals |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-4e612bddf6b44eec854bd398ee72c6572025-08-20T02:38:48ZengWileyAdvances in Mathematical Physics1687-91201687-91392016-01-01201610.1155/2016/70580177058017The Ritz Method for Boundary Problems with Essential Conditions as ConstraintsVojin Jovanovic0Sergiy Koshkin1Systems, Implementation & Integration, Smith Bits, A Schlumberger Co., 1310 Rankin Road, Houston, TX 77032, USAComputer and Mathematical Sciences, University of Houston-Downtown, One Main Street #S705, Houston, TX 77002, USAWe give an elementary derivation of an extension of the Ritz method to trial functions that do not satisfy essential boundary conditions. As in the Babuška-Brezzi approach boundary conditions are treated as variational constraints and Lagrange multipliers are used to remove them. However, we avoid the saddle point reformulation of the problem and therefore do not have to deal with the Babuška-Brezzi inf-sup condition. In higher dimensions boundary weights are used to approximate the boundary conditions, and the assumptions in our convergence proof are stated in terms of completeness of the trial functions and of the boundary weights. These assumptions are much more straightforward to verify than the Babuška-Brezzi condition. We also discuss limitations of the method and implementation issues that follow from our analysis and examine a number of examples, both analytic and numerical.http://dx.doi.org/10.1155/2016/7058017 |
| spellingShingle | Vojin Jovanovic Sergiy Koshkin The Ritz Method for Boundary Problems with Essential Conditions as Constraints Advances in Mathematical Physics |
| title | The Ritz Method for Boundary Problems with Essential Conditions as Constraints |
| title_full | The Ritz Method for Boundary Problems with Essential Conditions as Constraints |
| title_fullStr | The Ritz Method for Boundary Problems with Essential Conditions as Constraints |
| title_full_unstemmed | The Ritz Method for Boundary Problems with Essential Conditions as Constraints |
| title_short | The Ritz Method for Boundary Problems with Essential Conditions as Constraints |
| title_sort | ritz method for boundary problems with essential conditions as constraints |
| url | http://dx.doi.org/10.1155/2016/7058017 |
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