Mathematical theory of the finite volume methods

2014 Fiscal Year Final Research Report

• Project/Area Number: 23340023
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: The University of Tokyo
• Principal Investigator: SAITO Norikazu 東京大学, 数理(科)学研究科(研究院), 准教授 (00334706)
• Co-Investigator(Kenkyū-buntansha): TSUCHIYA Takuya 愛媛大学, 大学院理工学研究科, 教授 (00163832) ; YAGUCHI Masaharu 神戸大学, 大学院システム情報学研究科, 講師 (10396822) ; FURIHATA Daisuke 大阪大学, サイバーメディアセンター, 准教授 (80242014) ; MURAKAWA Hideki 九州大学, 大学院数理学研究科, 助教 (40432116)
• Co-Investigator(Renkei-kenkyūsha): KIKUCHI Fumio 東京大学, 名誉教授 (40013734) ; KAWARADA Hideo 千葉大学, 名誉教授 (90010793) ; USHIJIMA Teruo 電気通信大学, 名誉教授 (10012410)
• Research Collaborator: MIYASHITA Masaru
• Project Period (FY): 2011-04-01 – 2015-03-31
• Keywords: 数値解析 / 数理モデル / 有限体積法 / 有限要素法 / 差分法

Outline of Final Research Achievements

This research project was aimed at development and application of the mathematical theory for the finite volume method that is a popular structure-preserving discretization method. From the mathematical stand-point, the discrete Sobolev inequality, interpolation error constants, discrete Rellich's theorem, discrete maximum principle, and discrete differential form were studied and many useful results were obtained. As an important application, results were applied to analysis of the finite volume method for the mathematical model describes the aggregation of slime molds resulting from their chemotactic features. In particular, the proof of the existence of a discrete free energy was succeeded. Another important application was an extension of energy-preserving numerical method based on Lagrange mechanics to the finite volume method by using the theory of the discrete differential form.

Free Research Field

数値解析