2006 Fiscal Year Final Research Report Summary
Synthetic approach for the development of computer assisted analysis from the numerical verification methods
| Project/Area Number |
15204007
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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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 UNIVERCITY |
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Principal Investigator |
NAKAO Mitsuhiro Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (10136418)
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| Co-Investigator(Kenkyū-buntansha) |
TABATA Masahisa Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (30093272)
IMAI Hitoshi Tokushima University, Faculty of Engineering, Professor, 工学部, 教授 (80203298)
TSUCHIYA Takuya Ehime University, Faculty of Science, Professor, 理学部, 教授 (00163832)
NISHIDA Takaaki Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (70026110)
CHIN Shokun Hirosaki University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70304251)
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| Project Period (FY) |
2003 – 2006
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| Keywords | Numerical analysis / Validated computation / Numerical verification / Computer assisted proof / 計算機援用証明 |
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| Research Abstract |
In this research, we newly developed the numerical verification methods which can be applied to wide mathematical and analytical problems, as well as extended or improved the existing techniques.
And we actually applied these methods to particular problems such as equations in the mathematical fluid mechanics and oscillation problems etc. The important research results obtained by investigators and co-investigators are as follows :
1.Nakao, N.Yamamoto, Watanabe established several refinements and extensions for the constructive error estimates for the finite finite element projections of the Poisson and the bi-harmonic equations on various kinds of domains, particularly, on nonconvex polygonal domains. These results played important and essential roles for the numerical verification of solutions of nonlinear elliptic equations and the two dimensional stationary Navier-Stokes problems.
2.Nagatou numerically proved the stability of the flow on the torus called Kolmogorov problem.
3.Minamoto presented a formulation of the verification condition for the double turning point and applied it to the perturbed Gelfand equation.
4.Oishi established some refinements on the fast algorithm for the solutions of linear equations.
5.Nishida et al. presented the computed results with guaranteed error bounds for the symmetry breaking bifurcation point of the solution of two dimensional heat convection problems, as well as they formulated the numerical verification algorithm for the three dimensional problems with some prototypical verified examples.
6.Chin obtained some numerical verification results on the existence of solutions and a posteriori error estimates for the linear complementarity problems.
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