Fundamentals of Mathematical Science for Industrial Applications of Discrete Geometric Mechanics

Reference No.	2022a028
Type/Category	Grant for General Research-Short-term Joint Research
Title of Research Project	Fundamentals of Mathematical Science for Industrial Applications of Discrete Geometric Mechanics
Principal Investigator	Hirotoshi Kuroda（Department of Mathematics, Faculty of Science, Hokkaido University・Associate Professor）
Research Period	January 21,2023. ～ January 22,2023.
Keyword(s) of Research Fields	Discrete Geometric Mechanics, Finite Element Exterior Calculus, Structure-Preserving Numerical Methods, Numerical Analysis
Abstract for Research Report	The finite element exterior calculus (FEEC) was proposed around 2006 by Aronold-Falk-Winther et al. This is a unified method for constructing adaptive and stable mixed finite element schemes for certain elliptic equations and providing error evaluations for them. In recent years, many attempts have been made to incorporate differential forms into mathematical models of physical phenomena and numerical simulations of partial differential equations. By treating ``flow'' in differential form, distinguishing between flow (as 1-form) and flux (as 2-form), the relationship between space and physical quantities becomes clearer, and it is also useful tool for treating the results of observations through instruments that we can actually observe in a mathematical model. This concept has been adopted in engineering and excellent numerical solution methods have been developed. Although there are variations among different fields, it is called by various names such as Discrete External Differential Analysis, Discrete Differential Geometry, Difference Form, Vector Difference Analysis, and so on, depending on the field of application. In this research project, we will focus on the finite element exterior calculus as one of these topics. First, we will invite Associate Professor Taniguchi, who has related achievements, as a lecturer, and then attempting to understand the mathematical basis of this field. Next, we will work on the development of a numerical method that can preserve the structure of differential equations even when they are discretized by using finite element exterior calculus, with the subject of numerical calculations of electromagnetic field problems etc. that Associate Professor Tagami, one of the participants, has been working on. This is expected to lead to the proposal of numerical methods for electromagnetic field problems with time evolution based on finite element exterior calculus and the development of computer-assisted proofs through industrial applications of the methods. Furthermore, by penetrating the mathematical basis of the finite element exterior calculus, the possibility of its application to new fields, such as computer graphics and numerical calculations for various membrane structures, will be discussed.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Takaharu Yaguchi（Graduate School of System Informatics, Kobe University・Associate Professor） Jun Masamune（Mathematical Institute, Tohoku University・Professor） Hirotoshi Kuroda（Faculty of Science, Hokkaido University・Associate Professor） Takahiro Hasebe（Faculty of Science, Hokkaido University・Associate Professor） Hiroshi Teramoto（Faculty of Engineering Science, Kansai University・Associate Professor） Daisuke Tagami（Institute of Mathematics for Industry, Kyushu University・Associate Professor） Takanobu Hara（Faculty of Science, Hokkaido University・Postdoctoral Fellow） Yuhan Chen（Graduate School of System Informatics, Kobe University・D1） Shunpei Terakawa（Graduate School of System Informatics, Kobe University・M2） Baige Xu（Graduate School of System Informatics, Kobe University・M1）