Construction of fundamental theory of difference method based on discrete function analysis and variational theory

2017 Fiscal Year Final Research Report

• Project/Area Number: 25287030
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Osaka University
• Principal Investigator: Furihata Daisuke 大阪大学, サイバーメディアセンター, 教授 (80242014)
• Co-Investigator(Kenkyū-buntansha): 松尾 宇泰 東京大学, 大学院情報理工学系研究科, 教授 (90293670)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Keywords: 構造保存数値解法 / 偏微分方程式 / 差分法 / 変分理論 / 離散変分導関数法

Outline of Final Research Achievements

We have studied the discrete function analysis and the discrete variational theory. We proposed the concept of the discrete transformation matrix in the space defined via difference operators based on discrete calculus between differential operators and integral ones and evaluated those mathematically.Based on these results, we proved some discrete differential-integral inequalities in general and studied mathematical conditions to satisfy them.We have deepened the understanding of mathematical constraints by discussing the proof technique and gained a rigorous suggestion that this argument holds in wider function spaces.
We also promoted research to extend the discrete definitions of key concepts used in discrete variational theory.As a result, the discrete variational theory concept was extended based on the discrete version of the basic relational expression such as the Green theory, and the further mathematical basis of the difference method was defined.

Free Research Field

微分方程式の構造保存数値解法