Self-validating numerics with applications to computational science and technology

自验证数值及其在计算科学和技术中的应用

• 批准号: 10440031
• 负责人: KANAO Maitsuhiro
• 金额: $ 7.62万
• 依托单位: KYUSHU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10440031/
• 关键词: Numerical analysis; Validated numerical computation; Numerical verification method; Computer assisted proof

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 … More 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

在本研究中，我们对自验证数值方法进行了扩展和改进，该方法不仅可以应用于广泛的数学和分析问题，也可以应用于数学流体力学中的方程等特殊问题。研究者和合作研究者所做的重要研究成果如下：1。（by Nakao， N. Yamamoto和Watanabe）建立了椭圆型问题解的数值验证方法的若干改进和推广。利用非线性椭圆边值问题解数值验证方法中的技术，建立了二阶椭圆算子特征值问题具有保证误差界的数值计算方法。给出了椭圆型变分不等式解的自验证方法，并得到了基本结果。此外，我们给出了基于a后验的Navier-Stokes方程解的验证计算和Stokes问题有限元解的构造性先验误差估计。此外，我们还计算了扰动参数化Gelfand方程具有严格误差界的拐点。（by Oishi）给出了一些用于基本验证计算和求解线性和非线性问题的快速算法。对一类特殊的电磁问题有限元方法进行了误差分析，得到了理论和数值结果。（by Sakai）介绍了样条曲线在平面数据逼近中的一些应用。（by Fujino）研究了一种有效的并联机构加速方法。（by Mitsui）提出一种具有初值问题的常微分方程的自验证方法。对Dirichlet问题的Shortley-Weller型差分格式进行了一些新的误差分析。对流体力学问题的有限元法给出了几个误差估计。（by Nishida）用计算机辅助证明方法分析了流体力学中的一些分岔现象。利用群理论分岔论证，研究了结构工程中的可靠度问题。少

项目成果

Watanabe, Y.: "A numerical verfication method of solutions for the Navier-Stokes equations"Reliable Computing. 5. 347-357 (1999)

Watanabe, Y.：“纳维-斯托克斯方程解的数值验证方法”可靠计算。

Nakao, M.T.: "A posteriori constructive a priori error bounds for finite element solutions of Stokes equations"Journal of Computational and Applied Mathematics. 91. 137-158 (1998)

Nakao，M.T.：“斯托克斯方程有限元解的后验构造先验误差界”计算与应用数学杂志。

Feng, B. F.: "A conservative spectral method for the third- and fifth-order Korteweg-de Vries Equations"Journal of Computational Physics. 153. 467-487 (1999)

Feng, B. F.：“三阶和五阶 Korteweg-de Vries 方程的保守谱方法”计算物理学杂志。

Feng,B.-F.: "A conservative spectral method for the third-and fofth-order Kortewege-de Vries equations"Journal of Computational Physics. 153. 467-487 (1999)

Feng,B.-F.：“三阶和四阶 Kortewege-de Vries 方程的保守谱方法”计算物理学杂志。

Nakao,M.T.: "Numerical verifications of eigenvalues of second-order elliptic operators"Japan Journal of Industrial and Applied Mathematics. 16. 307-320 (1999)

Nakao,M.T.：“二阶椭圆算子特征值的数值验证”日本工业与应用数学杂志。