Accurate and efficient numerical solutions for elliptic obstacle problems
Abstract Elliptic obstacle problems are formulated to find either superharmonic solutions or minimal surfaces that lie on or over the obstacles, by incorporating inequality constraints. In order to solve such problems effectively using finite difference (FD) methods, the article investigates simple...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-02-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-017-1309-z |
| _version_ | 1819057415701659648 |
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| author | Philku Lee Tai Wan Kim Seongjai Kim |
| author_facet | Philku Lee Tai Wan Kim Seongjai Kim |
| author_sort | Philku Lee |
| collection | DOAJ |
| description | Abstract Elliptic obstacle problems are formulated to find either superharmonic solutions or minimal surfaces that lie on or over the obstacles, by incorporating inequality constraints. In order to solve such problems effectively using finite difference (FD) methods, the article investigates simple iterative algorithms based on the successive over-relaxation (SOR) method. It introduces subgrid FD methods to reduce the accuracy deterioration occurring near the free boundary when the mesh grid does not match with the free boundary. For nonlinear obstacle problems, a method of gradient-weighting is introduced to solve the problem more conveniently and efficiently. The iterative algorithm is analyzed for convergence for both linear and nonlinear obstacle problems. An effective strategy is also suggested to find the optimal relaxation parameter. It has been numerically verified that the resulting obstacle SOR iteration with the optimal parameter converges about one order faster than state-of-the-art methods and the subgrid FD methods reduce numerical errors by one order of magnitude, for most cases. Various numerical examples are given to verify the claim. |
| first_indexed | 2024-12-21T13:38:57Z |
| format | Article |
| id | doaj.art-1a277326451f4937a812049a817036bb |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-21T13:38:57Z |
| publishDate | 2017-02-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-1a277326451f4937a812049a817036bb2022-12-21T19:02:05ZengSpringerOpenJournal of Inequalities and Applications1029-242X2017-02-012017112510.1186/s13660-017-1309-zAccurate and efficient numerical solutions for elliptic obstacle problemsPhilku Lee0Tai Wan Kim1Seongjai Kim2Department of Mathematics, Sogang UniversityCentennial Christian School InternationalMississippi State UniversityAbstract Elliptic obstacle problems are formulated to find either superharmonic solutions or minimal surfaces that lie on or over the obstacles, by incorporating inequality constraints. In order to solve such problems effectively using finite difference (FD) methods, the article investigates simple iterative algorithms based on the successive over-relaxation (SOR) method. It introduces subgrid FD methods to reduce the accuracy deterioration occurring near the free boundary when the mesh grid does not match with the free boundary. For nonlinear obstacle problems, a method of gradient-weighting is introduced to solve the problem more conveniently and efficiently. The iterative algorithm is analyzed for convergence for both linear and nonlinear obstacle problems. An effective strategy is also suggested to find the optimal relaxation parameter. It has been numerically verified that the resulting obstacle SOR iteration with the optimal parameter converges about one order faster than state-of-the-art methods and the subgrid FD methods reduce numerical errors by one order of magnitude, for most cases. Various numerical examples are given to verify the claim.http://link.springer.com/article/10.1186/s13660-017-1309-zelliptic obstacle problemsuccessive over-relaxation (SOR) methodgradient-weighting methodobstacle relaxationsubgrid finite difference (FD) |
| spellingShingle | Philku Lee Tai Wan Kim Seongjai Kim Accurate and efficient numerical solutions for elliptic obstacle problems Journal of Inequalities and Applications elliptic obstacle problem successive over-relaxation (SOR) method gradient-weighting method obstacle relaxation subgrid finite difference (FD) |
| title | Accurate and efficient numerical solutions for elliptic obstacle problems |
| title_full | Accurate and efficient numerical solutions for elliptic obstacle problems |
| title_fullStr | Accurate and efficient numerical solutions for elliptic obstacle problems |
| title_full_unstemmed | Accurate and efficient numerical solutions for elliptic obstacle problems |
| title_short | Accurate and efficient numerical solutions for elliptic obstacle problems |
| title_sort | accurate and efficient numerical solutions for elliptic obstacle problems |
| topic | elliptic obstacle problem successive over-relaxation (SOR) method gradient-weighting method obstacle relaxation subgrid finite difference (FD) |
| url | http://link.springer.com/article/10.1186/s13660-017-1309-z |
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