High-precision eigenvalue estimation for the Biharmonic differential operator

2018 Fiscal Year Final Research Report

• Project/Area Number: 26800090
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Niigata University
• Principal Investigator: LIU Xuefeng 新潟大学, 自然科学系, 准教授 (50571220)
• Project Period (FY): 2014-04-01 – 2019-03-31
• Keywords: 微分作用素の固有値問題 / 有限要素法 / 誤差評価理論 / 精度保証付き数値計算 / 重調和微分作用素 / Stokes微分作用素 / Steklov微分作用素

Outline of Final Research Achievements

To give lower and upper bounds for the eigenvalues of differential operators is one of the fundamental problems in the history of mathematics. On the opposite side of easy-to-obtain upper eigenvalue bounds, it is difficult to provide lower eigenvalue bounds for the operators. In this research, the researcher proposed a general framework to bound eigenvalues of differential operators, which can be performed along with the conforming finite element method (FEM) or the non-conforming one. Such a framework has been successfully applied to provide explicit bounds for the eigenvalues of the Biharmonic operator, the Stokes operator and the Steklov operator. One feature of proposed eigenvalue estimation is that, it takes the advantages of the nice property of special non-conforming finite element methods, such as the Crouzeix-Raviart FEM, the Fujino-Morley FEM, to give concise and efficient lower eigenvalue bounds evaluation.

Free Research Field

数学基礎・応用数学

Academic Significance and Societal Importance of the Research Achievements

微分作用素の厳密な固有値評価は非線形方程式の解の計算機援用存在証明などの研究に重要な役割を果たしています。本研究で提案した固有値の評価方法によって多くの微分作用素の厳密な固有値評価が可能となり、当該方法が数値計算の品質保証や計算機援用証明の難題解決に貢献できると思われます。