Study of finite element methods for nonlinear problems and its error analysis

1.我们研究了自由边界问题，称为渗流问题或大坝问题。无论如何，本文采用了一些水平集方法对大坝问题进行了理论分析。然而，在工程中通常采用迭代格式来计算数值解。由于迭代格式的严格分析是非常困难的，关于大坝问题的迭代格式的数学结果很少。在我们的研究中，我们为大坝问题的数值迭代方法的收敛分析提供了一个数学框架(见[1，3])。要为自由边界问题设计“好的”迭代格式，重要的是要了解当域的边界被扰动时，与问题有关的量将如何变化。这种变种被称为阿达玛变种。在我们的研究中，我们成功地计算了速度势关于边界摄动的第一变分。T.Suzuki，T.T.Tuchiya，《Hadamard变分的弱公式及其在过滤问题中的应用》，预印本。分析了分段二次有限元方法在两点边值问题中的应用。我们用的是“山本原理”。由于山本原理是一个强大的工具，我们可以处理标准理论无法处理的情况。我们证明，即使系数函数仅是分段光滑的，所有标准结果仍然有效(见[3])。

1. We have studied on the free boundary problem called the filtration problem or the dam problem. In any cases, some level-set approaches are taken to analyze the dam problem theoretically. However, to compute the numerical solutions, iterative schemes are used in engineering usually. Since rigorous analysis of iterative schemes is very difficult, there are very few mathematical results on iterative schemes for the dam problem. In our study, we present an mathematical framework for convergence analysis of numerical iterative methods for the dam problem (see [1,3]).2. To design "good" iterative scheme for free boundary problems, it is important to understand how quantities related to the problem will vary when the boundary of the domain is perturbed. Such a variation is called the Hadamard variation. In our study, we have succeeded to compute the first variation of the velocity potential with respect to boundary perturbation.T. Suzuki, T. Tsuchiya, "Weak formulation of Hadamard variation and its application to the filtration problem", preprint.3. We analyze the piecewise quadratic finite element method applied to 2-point boundary value problems. We use "Yamamoto's principle" for it. Since Yamamoto's principle is a powerful tool, we can deal with cases which the standard theory cannot handle. We conform that all standard results are still valid even if coefficient functions are only piecewise smooth (see [3]).