Self-validating numerics with applications to computational science and technology

在这项研究中，我们扩展和改进的自验证数值方法，可以适用于广泛的数学和分析问题，以及特定的问题，如方程的数学流体力学。研究者和合作者所做的重要研究成果如下：1。(by Nakao，N. Yamamoto和Watanabe）对椭圆问题解的数值验证方法进行了改进和推广。即利用非线性椭圆边值问题解的数值验证方法中的技巧，建立了二阶椭圆算子特征值问题具有保误差界的数值计算方法.我们还制定和椭圆型变分不等式的解决方案的自验证方法的基本结果。此外，我们还提出了一个基于后验的Navier-Stokes方程解的验证计算方法 ...更多信息 i和Stokes问题有限元解的构造性先验误差估计。此外，我们还计算了摄动参数化Gelfand方程的一个具有严格误差界的转向点. (by Oishi）提出了一些基本验证计算和线性与非线性问题求解的快速算法. (by菊池）对电磁场问题中一类特殊的有限元方法进行了误差分析，得到了理论和数值结果. (by介绍了样条函数在平面数据逼近中的一些应用. (by研究了一种高效的并行机加速方法. (by Mitsui）提出了一种求解常微分方程初值问题的自验证方法. (by T. Yamamoto）对Dirichlet问题的Shortley-Weller型差分格式进行了新的误差分析. (by Tabata）对流体力学中的问题导出了有限元法的几个误差估计. (by Nishida）对流体动力学中的一些分岔现象进行了计算机辅助证明. (by Murota）利用群论分歧理论研究了结构工程中的可靠性问题。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