Numerical verification method of solutions for nonlinear evolutional equations

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540151
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Sasebo National College of Technology
• Principal Investigator: NAKAO Mitsuhiro 佐世保工業高等専門学校, その他部局等, その他 (10136418)
• Co-Investigator(Kenkyū-buntansha): KIMURA Takuma 早稲田大学, 理工学術院, 次席研究員 (60581618)
• Co-Investigator(Renkei-kenkyūsha): WATANABE Yoshitaka 九州大学, 情報基盤研究開発センター, 准教授 (90243972) ; KINOSHITA Takehiko 京都大学, 数理解析研究所, 研究員 (30546429)
• Research Collaborator: MICHAEL Plum Karlsruhe大学(ドイツ), 教授
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 精度保証付き数値計算法 / 解の数値的検証 / 解の事後誤差評価 / 非線形発展方程式 / 計算機援用証明

Outline of Final Research Achievements

We studied the numerical verification of solutions for nonlinear evolutional problems with some numerical examples. We considered the constructive a priori error estimates for a full discrete numerical solution of the heat equation, which is based on the finite element Galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space. We also considered the guaranteed a posteriori estimates for the inverse parabolic operators and derived a numerical verification method of exact solutions for the nonlinear parabolic equations by Newton's method with some prototype numerical examples. Furthermore, we presented some refinements on the verified computations for linearlized inverse operators of elliptic operators as well as the eigenvalue excluding techniques for them.

Free Research Field

計算数学