Mathematical Theory of Error Analysis of Finite Element Methods

在本研究计画中，我们尝试发展有限元素法误差分析的数学理论，并取得下列结果。(1)We考虑了Hilbert空间中带紧项（线性或非线性）的（非）线性方程组的Galerkin方法的误差分析。我们已经找到了一个简单的方法来发展数学理论的误差分析的伽辽金方法应用于这样的方程。(2)We严格证明了试自由边界法应用于大坝问题这一典型的椭圆型自由边界问题的收敛性。(3)We考虑了一维有界区间上两点边值问题的分片二次有限元方法的误差分析。我们发现，即使主项的系数函数不连续或完全为正，所有已知的误差界仍然有效。

In this research project, we have tried to develop a mathematical theory of error analysis of finite element methods, and have obtained the following results.(1)We have consider on error analysis of Galerkin methods applied to (non-) linear equations with a (linear or nonlinear) compact term defined in Hilbert space setting. We have found a simple way of developing mathematical theory of error analysis of Galerkin methods applied to such equations.(2)We have proved rigorously the convergence of trial free boundary methods applied to the Dam Problem, which is a typical elliptic free boundary problem.(3)We have consider on error analysis of the piecewise quadratic finite element method applied to 2-point boundary value problems defined on 1-dimensional bounded interval. We have found that all known error bounds are still valid even if coefficient function of the principle term is not-continuous or entirely positive.