Study of interpolation error analysis for finite element methods

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400201
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Ehime University
• Principal Investigator: Tsuchiya Takuya 愛媛大学, 理工学研究科(理学系), 教授 (00163832)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 誤差解析 / 関数補間 / 有限要素法 / 外接半径

Outline of Final Research Achievements

In this study, we have considered the error analysis of Lagrange interpolation on triangles and tetrahedrons (hereafter, we call them "elements"). Usually, the error analysis of Lagrange interpolation is done under the condition that elements are not so degenerate (or not so “flat”). We have found that essential factor of the error estimation is the (projected) circumradius of elements, and presented new error estimations expressed in terms of the (projected) circumradius and the diameter of elements.
According to the newly obtained error estimation, we can say that if the (projected) circumradius of elements converges to 0, the error of Lagrange interpolations also converges to 0, even if elements are become very flat. Therefore, if we use triangulations with such elements, finite element methods would provide reliable numerical solutions.
We believe that the newly obtained error estimation provide a new insights to finite element methods and numerical simulations.

Free Research Field

数値解析学、偏微分方程式に対する数値解法の数学的基礎