A study on the numerical verification method of solutions with high accuracy for the nonlinear mathematical models in infinite dimension

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K05012
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kyushu University (2016-2017); Sasebo National College of Technology (2015)
• Principal Investigator: NAKAO Mitsuhiro 九州大学, マス・フォア・インダストリ研究所, 学術研究者 (10136418)
• 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) ; KIMURA Takuma 佐賀大学, 理工学部, 准教授 (60581618) ; KINOSHITA Takehiko 京都大学, 学際融合教育研究推進センター, 特定講師 (30546429)
• Research Collaborator: Michael Plum Karlsruhe大学, 教授
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 精度保証付き数値計算法 / 解の数値的検証 / 数値解析 / 解の事後誤差評価 / 計算機援用証明

Outline of Final Research Achievements

We studied the numerical verification method of solutions for infinite dimensional nonlinear mathematical models including elliptic and parabolic equations. We presented several computational techniques for the numerical estimations of the linearized inverse operators associated with nonlinear problems. From the viewpoint of efficiency and accuracy in the verified computations, we proposed several techniques which enebles us the actual effectiveness by showing numerical examples related to elliptic problems of second and fourth order. We also derived the constructive a priori error estimates with optimal order for a full discrete numerical scheme of the heat equation, which is based on the finite element Galerkin method with an interpolation in time using the fundamental matrix for ODEs. Furthermore, under the general setting in Hilbert space, we presented a principle of the verified computational method of solutions using Newton-type formulation.

Free Research Field

計算数学