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

1.我们研究的自由边界问题称为渗流问题或大坝问题。在任何情况下，一些水平集方法被用来分析大坝问题的理论。然而，在工程上通常采用迭代法来计算数值解。由于迭代格式的严格分析是非常困难的，因此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，我们提出了一个数学框架的收敛性分析的数值迭代方法的大坝问题（见[1，3]）。为了设计一个好的自由边界问题的迭代格式，重要的是要了解当区域边界受到扰动时，与问题有关的量将如何变化。这种变化被称为阿达玛变化。在我们的研究中，我们成功地计算了速度势关于边界扰动的第一变分。铃木，T. Tsuchiya，“弱阿达玛变分公式及其在过滤问题中的应用”，预印本. 3.分析了分片二次有限元法在两点边值问题中的应用。由于山本原理是一个强有力的工具，我们可以处理标准理论不能处理的情况。我们证明了即使系数函数是分段光滑的，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]).