1999 Fiscal Year Final Research Report Summary
Self-validating numerics with applications to computational science and technology
| Project/Area Number |
10440031
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | KYUSHU UNIVERSITY |
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Principal Investigator |
KANAO Maitsuhiro Kyushu University, Graduate school of Mathematics, Proffesor, 大学院・数理学研究科, 教授 (10136418)
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| Co-Investigator(Kenkyū-buntansha) |
MUROTA Kazuo Kyoto University, Res. Inst. Math. Sci., Professor, 数理解析研究所, 教授 (50134466)
NISHIDA Takaaki Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (70026110)
OISHI shin'ichi Waseda University, Informatics, Professor, 理工学部, 教授 (20139512)
YAMAMOTO Tetsuro Ehime University, Mathematics, Professor, 理学部, 教授 (80034560)
MITSUI Taketomo Nagoya University, Human Informatics, Professor, 大学院・人間情報学研究科, 教授 (50027380)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Numerical analysis / Validated numerical computation / Numerical verification method / Computer assisted proof |
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| Research Abstract |
In this research, we extended and improved the self-validating numerical methods which can be applied to wide mathematical and analytical problems as well as to particular problems such as equations in the mathematical fluid mechanics. The important research results done by investigators and co-investigators are as follows :
1. (by Nakao, N. Yamamoto and Watanabe) Several refinements and extensions were established for the numerical verification methods of solutions for elliptic problems. Namely, the numerical computation with guaranteed error bounds for the eigenvalue problems of second order elliptic operator was established by using the techniques in the numerical verification method of solutions for nonlinear elliptic boundary value problems. We also formulated and obtained basic results for the self-validating method for solutions of elliptic variational inequalities,. Moreover, we presented a verified computation of solutions for the Navier-Stokes equation based on the a posterior
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i and constructive a priori error estimates for the finite element solutions of the Stokes problems. Additionally, we computed a turning point with rigorous error bound for the perturbed and parameterized Gelfand equation.
2. (by Oishi) Some fast algorithms for the fundamental validated computations and the solutions of linear and nonlinear problems were presented.
3. (by Kikuchi) Theoretical and numerical results were obtained for the error analysis of a special kind of finite element method for electro-magnetic problems.
4. (by Sakai) Some applications of splines were presented for plane data approximation.
5. (by Fujino) An efficient acceleration method was investigated for parallel machines.
6. (by Mitsui) A self-validating method for ordinary differential equations with initial value problems was presented.
7. (by T. Yamamoto) Some new error analysis was carried out for the Shortley-Weller type deference scheme for Dirichlet Problems.
8. (by Tabata) Several error estimates were derived of the finite element method for the problem in fluid mechanics.
9. (by Nishida) Some bifurcation phenomena in fluid dynamics were analyzed by the computer assisted proof.
10. (by Murota) The reliability in the structural engineering was investigated by using the group theoretic bifurcation arguments. Less
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