Innovative research for self-validating numerical method of infinite dimensional problems
| Project/Area Number |
18K03434
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Waseda University |
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Principal Investigator |
NAKAO MITSUHIRO 早稲田大学, 理工学術院, その他(招聘研究員) (10136418)
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| Project Period (FY) |
2018-04-01 – 2022-03-31
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| Project Status |
Completed (Fiscal Year 2021)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
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| Keywords | 数値解析 / 精度保証付き数値計算 / 有限要素法の構成的誤差評価 / 解の数値的検証法 / 計算機援用証明 / 誤差評価 / 数値的検証 |
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| Outline of Final Research Achievements |
For infinite dimensional problems mainly centered on PDEs, we extended and improved the method of finding their solutions by numerical verificatioon methods. We presented efficient estimates of the norm for linearized inverse operator, which is important in the numerical verification of the solution of nonlinear elliptic problems, and also applied it to the non-existence proof of the solution. Moreover, based on the finite element approximation and the error estimates, we realized the guranteed computation for the solution of the stationary Navier-Stokes equation in the three-dimensional general domain, which was difficult in the past. Also by giving constructive error estimates for the semi-discrete solution of the heat equation, we improved the verification efficiency of the solution for parabolic problems. Regarding the blow-up solution of the nonlinear evolution equation, we formulated the guaranteed computation of the blow-up time with a numerical example.
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| Academic Significance and Societal Importance of the Research Achievements |
近年の計算機技術の進歩によって、偏微分方程式を含めた非線形数理モデルに対する計算機援用証明(数値的検証)は、現象の理論的解明において重要な手段となりつつある。しかしながら、特に偏微分方程式の場合には、その誤差評価が複雑で精度も不十分なために適用対象が限定され、応用解析学や計算理工学上に現れる多くの実際的非線形問題に対し、その実用性は未だに高いとは言い難い。本研究は、そのような難点を克服する手法の開発を目ざして遂行したものである。本研究の成果では、流体問題の数値シミュレーションに対する信頼性保証に成功するなど、この分野のさらなる発展についてその突破口を見いだすことができた。
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Report
(5 results)
Research Products
(43 results)
