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

本项目根据需要对Banach空间牛顿法的收敛定理进行了重新表述和优化，开展了微分方程数值分析的理论研究。更确切地说，我们建立了一个有效的算法的数值验证的非线性偏微分方程的解决方案，这是基于一个新的简化收敛定理的牛顿法。我们澄清了一些验证的例子，我们的方法是更有效的验证解决方案比其他已知的方法。牛顿法的收敛性定理在原理上是明确的，从理论上讲是非常优秀的。同时，长期以来，相关研究者认为该定理从计算效率的角度来看并不好，因此不太适合于偏微分方程解的验证。我们一定会以我们的成就来推翻他们的修复概念。我们的论文包括上述结果发表在J. COMPUT。在基于有限元方法的数值验证中优化了上述收敛定理。然而，从计算精度的观点来看，有限元方法通常较差，并且不太适合于对诸如动力系统中的分叉的复杂现象进行精确分析。谱方法就是谱方法。此外，我们还推广了线性化算子逆的范数估计方法（这是检验牛顿法收敛性定理中一个条件的重要规则），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.