Synthetic approach for the development of computer assisted analysis from the numerical verification methods

从数值验证方法发展计算机辅助分析的综合方法

• 批准号: 15204007
• 负责人: NAKAO Mitsuhiro
• 金额: $ 20.63万
• 依托单位: KYUSHU UNIVERCITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15204007/
• 关键词: Numerical analysis; Validated computation; Numerical verification; Computer assisted proof; 計算機援用証明

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.

在本研究中，我们发展了适用于广泛的数学和分析问题的数值验证方法，扩展或改进了现有的技术，并将这些方法实际应用于数学流体力学方程和振动问题等特殊问题。研究者和合作者取得的重要研究成果如下：1.Nakao，N.Yamamoto，Watanabe建立了Poisson方程和双调和方程在各种区域上，特别是在非凸多边形区域上的有限元投影的构造性误差估计的若干改进和推广。这些结果对于非线性椭圆型方程和二维定常Navier方程解的数值验证具有重要意义。Stokes问题; 2. Nagatou数值证明了环面上流动的稳定性，称为Kolmogorov问题; 3. Minamoto提出了双转向点验证条件的公式，并将其应用于扰动Gelfand方程; 4. Oishi建立了一些改进的线性方程组解的快速算法。5. Nishida等人给出了二维热对流问题解的对称破缺分歧点的有保证误差界的计算结果，6. Chin对三维问题的解的存在性进行了数值验证，并给出了一些典型的验证实例线性互补问题的后验误差估计

项目成果

Numerical Verification of Solutions of Nekrasov's Integral Equation

Nekrasov积分方程解的数值验证

• 发表时间: 2005
• 期刊: Computing 75
• 作者: Murashige;S.
• 通讯作者: S.

Some computer assistedproofs for solutions of the heat convectionproblems

解决热对流问题的一些计算机辅助证明

• 发表时间: 2003
• 期刊: Reliable Computing Vol.9
• 作者: M.T.Nakao;Y.Watanabe;N.Yamamoto;T.Nishida
• 通讯作者: T.Nishida

Ryoo, C-S.: "Numerical verification of solutions for obstacle problems"Journal of computational and Applied Mathematics. 161. 405-416 (2003)

Ryoo, C-S.：“障碍问题解决方案的数值验证”计算与应用数学杂志。

Nakao, M.T.: "An efficient approach to the numerical verification for solutions of elliptic differential equations"Numerical Algorithms, Special issue for Proceedings of SCAN2002. (掲載決定).

Nakao, M.T.：“椭圆微分方程解的数值验证的有效方法”，数值算法，SCAN2002 论文集特刊（已出版）。

A computational approach to constructive a priori and a posteriori error estimates for finite element approximations of bi-harmonic problems

双调和问题有限元近似的构造性先验和后验误差估计的计算方法

• 期刊: GAKUTO International Series, Mathematical Sciences and Applications, Proceedings of the 4th JSIAM-SIMAI Seminar on Industrial and Applied Mathematics, May 26-28,2005,Hayama, Japan (to appear)
• 作者: Nakao;M.T.
• 通讯作者: M.T.