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

基本信息

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

项目摘要

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.

在这项研究中，我们新开发了可应用于广泛的数学和分析问题的数值验证方法，并扩展或改进了现有技术。并且我们实际上将这些方法应用于特定问题，例如数学流体力学中的方程和研究者和合作研究者取得的重要研究成果如下： 1.Nakao、N.Yamamoto、Watanabe 对泊松和有限元投影的构造误差估计进行了一些改进和扩展。各种域上的双调和方程，特别是非凸多边形域上的双调和方程。这些结果对于非线性椭圆方程和二维平稳纳维-斯托克斯问题的解的数值验证起到了重要和必要的作用。2.Nagatou数值证明了称为Kolmogorov问题的环面上流动的稳定性。3.Minamoto提出了一个公式的双转折点的验证条件并将其应用于扰动Gelfand方程。4.Oishi对线性方程组的快速求解算法进行了一些改进方程.5.Nishida 等人。给出了二维热对流问题解的对称性破缺分岔点的有保证误差界的计算结果，并通过一些原型验证算例制定了三维问题的数值验证算法。6.Chin 得到了一些数值验证结果。线性互补问题解的存在性和后验误差估计的验证结果。