Theoretical research on the numerical analysis for differential equations based on the convergence theorem of Newton's method

在这个项目中，我们根据需要对牛顿法巴拿赫空间的收敛定理进行了重新表述和优化，对微分方程的数值分析进行了理论研究。更精确地说，我们建立了一种基于新的简化牛顿法收敛定理的非线性偏微分方程解的数值验证算法。我们通过一些验证实例说明，我们的方法在验证解时比其他已知方法更有效。牛顿方法的收敛定理在原理上是明确的，从理论的角度来看是非常优秀的。与此同时，长期以来有关研究者认为，从计算效率的角度来看，该定理是不好的，因此它不太适合于偏微分方程解的验证。我们一定会用我们的成就来推翻他们的固定概念。我们包含上述结果的论文发表在J. Comput。达成。数学。在基于有限元方法的数值验证中，对上述收敛定理进行了优化。然而，从计算精度来看，有限元方法普遍较差，不适合对动力系统中的分岔等复杂现象进行精确分析。谱法是谱法。此外，我们还推广了估计线性化算子逆范数的方法（这在牛顿方法收敛定理的检验条件中起着重要的作用），以便将其应用到谱方法中。我们在希腊举行的2006年国际数值分析与应用数学会议上报告了上述结果并发表了演讲。

In this project we carried out the theoretical research on the numerical analysis for differential equations by reformulating and optimizing the convergence theorem of Newton's method Banach spaces according to our need. To be more precise, we established an efficient algorithm on the numerical verification for the solutions of nonlinear partial differential equations, which is based in a new simplifies convergence theorem of Newton's method. We clarify by some verification examples that our method is more efficient in the verification for solutions than the other known methods. The convergence theorem of Newton's method is clear in principle and is very excellent from the theoretical view point. At the same time it is long believed by the related researchers that this theorem is not good from the view point of the computational efficiency and that therefore it is not well suited to the verification for solutions of partial differential equations. We are sure to override their fixes concept by our achievement. Our paper including the above results was published in J. Comput. Appl. Math.The above convergence theorem is optimized in the numerical verification based on the finite element methods. the finite element methods is, however, inferior in general from the view point of the computational accuracy and is not well suited to the precise analysis for the complicated phenomena such as the bifurcation in dynamical systems. The spectrum method is spectrum methods. Moreover, we generalized the method on estimating the norm of the inverse of linearized operators (which plays an important rule in checking a condition in the convergence theorem of Newton's method) in order to apply it to the spectrum methods. We reported the above results and delivered a lecture at International conference of numerical analysis and applied mathematics 2006 held at Greece.