Mathematical Theory of Error Analysis of Finite Element Methods

有限元方法误差分析的数学理论

• 批准号: 14540122
• 负责人: TSUCHIYA Takuya
• 金额: $ 1.86万
• 依托单位: Ehime University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540122/
• 关键词: finite elements; error analysis; Galerkin methods; Mathematical foundation; 自由環境問題; ダムの問題

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.

在本研究项目中，我们尝试发展了有限元方法误差分析的数学理论，并获得了以下结果：(1)我们考虑了Galerkin方法应用于Hilbert空间中定义了(线性或非线性)紧项的(非线性)线性方程的误差分析问题。我们找到了一条发展Galerkin方法误差分析数学理论的简单途径。(2)我们严格地证明了无试探边界方法应用于大坝问题这一典型的椭圆自由边界问题的收敛。(3)考虑了分段二次有限元方法应用于一维有界区间上的两点边值问题的误差分析。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。

项目成果

Fang, Tsuchiya, Yamamoto: "Finite difference finite element and finite volume methods applied to two-point boundary value problems"Journal of Computational and Applied Mathematics. 139. 9-19 (2002)

Fang、Tsuchiya、Yamamoto：“应用于两点边值问题的有限差分有限元和有限体积方法”计算与应用数学杂志。

Takuya Tsuchiya: "Finite element approximations of parametrized strongly nonlinear boundary value problems"Japan Journal of Industrial and Applied Mathematics. 19. 377-398 (2002)

Takuy​​a Tsuchiya：“参数化强非线性边值问题的有限元近似”日本工业与应用数学杂志。

Matsuzawa, Y., Suzuki, T., Tsuchiya, T.: "Finite element approximation of H-surfaces"Mathematics of Computation. 72. 607-617 (2003)

Matsuzawa, Y.、Suzuki, T.、Tsuchiya, T.：“H 曲面的有限元近似”计算数学。

Takuya Tsuchiya: "Precise finite element error analysis by Yamamoto s explicit inversion formula for tridiagonal matrices An extension of Babuska-Osborn's theorems"Numerische Mathematik. 94. 541-572 (2003)

Takuy​​a Tsuchiya：“通过山本显式三对角矩阵反演公式进行精确有限元误差分析，巴布斯卡-奥斯本定理的扩展”数值数学。

T.Tsuchiya, K.Yoshida, S.Ishioka: "Yamamoto's principle and its applications to precise finite element error analysis"J.Comp.Appl.Math.. 152. 507-532 (2003)

T.Tsuchiya、K.Yoshida、S.Ishioka：“Yamamoto 原理及其在精确有限元误差分析中的应用”J.Comp.Appl.Math.. 152. 507-532 (2003)