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

在这项研究中，我们开发了新的数值验证方法，可以应用于广泛的数学和分析问题，并扩展或改进了现有的技术。我们实际上把这些方法应用到一些特殊的问题上，比如数学流体力学方程和振动问题等等。研究者及合作研究者取得的重要研究成果如下：1.研究成果；Nakao， N.Yamamoto， Watanabe对泊松方程和双调和方程在各种域上，特别是在非凸多边形域上的有限有限元投影的建设性误差估计建立了若干改进和扩展。这些结果对于非线性椭圆方程解的数值验证和二维平稳Navier-Stokes问题的数值验证具有重要的意义。Nagatou用数值方法证明了环面上流动的稳定性，称为Kolmogorov问题。Minamoto给出了双拐点验证条件的公式，并将其应用于摄动Gelfand方程。Oishi对求解线性方程的快速算法进行了改进。Nishida等人给出了二维热对流问题解的对称破缺分岔点有保证误差界的计算结果，并通过一些原型验证实例，制定了三维问题的数值验证算法。得到了线性互补问题解存在性的数值验证结果和后验误差估计。

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.