Self-validating numerics with applications to computational science and technology

在这项研究中，我们扩展和改进了自验证数值方法，它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。1.(Nakao，N.Yamamoto和Watanabe)对椭圆问题解的数值验证方法进行了几个改进和推广。利用非线性椭圆型边值问题解的数值验证方法中的技巧，建立了二阶椭圆型算子本征值问题的有保证误差界的数值计算。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，我们还给出了基于后验…的N-S方程解的验证计算。关于Stokes问题有限元解的更多先验误差估计。此外，我们还计算了摄动的参数Gelfand方程的一个具有严格误差界的转折点。(作者：Oishi)给出了一些基本验证计算和线性和非线性问题的快速算法。给出了电磁场问题一种特殊的有限元方法误差分析的理论和数值结果。(Sakai)给出了样条法在平面数据逼近中的一些应用。(Fujino)研究了一种用于并联机床的有效加速方法。给出了常微分方程组初值问题的一种自验证方法。T.Yamamoto)对Dirichlet问题的Shortley-Weller型差分格式进行了一些新的误差分析。(作者Tabata)导出了流体力学问题的有限元方法的几个误差估计。用计算机辅助证明对流体力学中的一些分叉现象进行了分析。(作者：Murota)用群论分叉方法研究了结构工程中的可靠性问题。较少

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