2006 Fiscal Year Final Research Report Summary
Study of finite element methods for nonlinear problems and its error analysis
| Project/Area Number |
17540120
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Ehime University |
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Principal Investigator |
TSUCHIYA Takuya Ehime University, Graduate School of Science and Engineering, Professor, 理工学研究科, 教授 (00163832)
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| Co-Investigator(Kenkyū-buntansha) |
SUZUKI Takashi Osaka University, Graduate School of Engineering Science., Professor, 基礎工学研究科, 教授 (40114516)
SAKAGUCHI Shigeru Ehime University, Graduate School of Science and Engineering, Professor, 理工学研究科, 教授 (50215620)
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| Project Period (FY) |
2005 – 2006
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| Keywords | finite element method / free boundary problem / error analysis / iterative scheme / Hadamard variation |
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| Research Abstract |
1. We have studied on the free boundary problem called the filtration problem or the dam problem. In any cases, some level-set approaches are taken to analyze the dam problem theoretically. However, to compute the numerical solutions, iterative schemes are used in engineering usually. Since rigorous analysis of iterative schemes is very difficult, there are very few mathematical results on iterative schemes for the dam problem. In our study, we present an mathematical framework for convergence analysis of numerical iterative methods for the dam problem (see [1,3]).
2. To design "good" iterative scheme for free boundary problems, it is important to understand how quantities related to the problem will vary when the boundary of the domain is perturbed. Such a variation is called the Hadamard variation. In our study, we have succeeded to compute the first variation of the velocity potential with respect to boundary perturbation.
T. Suzuki, T. Tsuchiya, "Weak formulation of Hadamard variation and its application to the filtration problem", preprint.
3. We analyze the piecewise quadratic finite element method applied to 2-point boundary value problems. We use "Yamamoto's principle" for it. Since Yamamoto's principle is a powerful tool, we can deal with cases which the standard theory cannot handle. We conform that all standard results are still valid even if coefficient functions are only piecewise smooth (see [3]).
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Research Products
(4 results)
