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

在本项目中，我们根据需要对牛顿法Banach空间的收敛定理进行了重新表述和优化，从而开展了微分方程组数值分析的理论研究。更准确地说，我们基于牛顿法的一个新的简化收敛定理，建立了一个有效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表明，我们的方法比其他已知的方法在解的验证上更有效。牛顿法的收敛定理在原理上是明确的，从理论上讲是非常优秀的。同时，相关研究人员也一直认为，该定理从计算效率的角度来看并不是很好，因此不适合于偏微分方程解的验证。我们一定会用我们的成就超越他们的修复理念。我们的论文包括了上述结果，发表在《计算机》杂志上。APPL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，从计算精度的角度来看，有限元方法总体上是较差的，不能很好地适应对动力系统中的分叉等复杂现象的精确分析。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(它在检验牛顿法收敛定理中的一个条件时起着重要的作用)，以便将其应用于谱方法。我们在希腊举行的2006年国际数值分析和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.