Researches on the high-accurate computation and numerical verification for the solution of the partial differential equation with singularity

• Project/Area Number: 22740059
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Hitotsubashi University (2011-2012); Kanazawa University (2010)
• Principal Investigator: KOBAYASHI Kenta 一橋大学, 大学院・商学研究科, 准教授 (60432902)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2010: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 精度保証付き数値計算 / 誤差評価 / 偏微分方程式 / 特異性 / 補間誤差 / 有限要素法 / 三角形要素 / 四面体要素 / 補間誤差評価 / 非線形方程式 / 補間誤差定数 / 流体方程式 / 解の存在と一意性 / 爆発解 / 精度保証 / 非凸領域 / メッシュリファインメント / 無限次元固有値問題

Research Abstract

The estimate for the interpolation error plays an essential role in error estimation for Finite Element Method. In our research, we obtained and proved precise formula that bounds interpolation error on the triangular elements. In particular, we proved that the interpolation error is bounded by the radius of circumscribed circle of the triangles. We call this condition circum radius condition. This result enables us to compute solutions of partial differential equation with singularity by efficient mesh division.

Research Products

• [Journal Article] Computer-AssistedUniqueness Proof for Stokes' Wave ofExtreme Form (2013)
  • Author(s): Kenta Kobayashi
  • Journal Title: Nankai Series in Pure, Applied Mathematics and Theoretical Physics Volume: 10巻 Pages: 54-67
• [Journal Article] Computer-Assisted Uniqueness Proof for Stokes' Wave of Extreme Form (2013)
  • Author(s): Kenta Kobayashi
  • Journal Title: Nankai Series in Pure, Applied Mathematics and Theoretical Physics Volume: 10 Pages: 54-67
• [Journal Article] 三角形要素上の補間誤差定数について (2011)
  • Author(s): 小林健太
  • Journal Title: 数理解析研究所講究録 Volume: 1733巻 Pages: 58-77
• [Journal Article] 三角形要素上の補間誤差定数について (2011)
  • Author(s): 小林健太
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1733 Pages: 58-77
• [Journal Article] On the global uniqueness of Stokes' wave of extreme form (2010)
  • Author(s): Kenta Kobayashi
  • Journal Title: IMA Journal of Applied Mathematics Volume: 75巻5号 Pages: 647-675
  • Peer Reviewed
• [Presentation] On the InterpolationConstants over Triangular Elements (2012)
  • Author(s): Kenta Kobayashi
  • Organizer: 2012 International Symposium on Nonlinear Theory and itsApplications(NOLTA2012)
  • Place of Presentation: Gran Melia Victoria,Palma,Majorca, Spain
  • Year and Date: 2012-10-26
• [Presentation] DE積分公式に対する使い勝手の良い事前誤差評価法 (2012)
  • Author(s): 小林健太
  • Organizer: 応用数理学会2012年度年会
  • Place of Presentation: 稚内全日空ホテル
  • Year and Date: 2012-08-31
• [Presentation] On the global uniqueness of Stokes' wave of extreme form (2011)
  • Author(s): Kenta Kobayashi
  • Organizer: Emerging Topics on Differential Equations and their Applications - Sino-Japan Conference of Young Mathematicians
  • Place of Presentation: Nankai University, China
  • Year and Date: 2011-12-06
• [Presentation] On the global uniqueness of Stokes' wave of extreme form (2011)
  • Author(s): Kenta Kobayashi
  • Organizer: Emerging Topics on Differential Equations and their Applications-Sino-Japan Conference of Young Mathematicians
  • Place of Presentation: Nankai University, Tianjin, China(招待講演)
  • Year and Date: 2011-12-06
• [Presentation] アメリカンオプションの最適権利行使境界に対する数値的検証法 (2011)
  • Author(s): 小林健太
  • Organizer: 日本応用数理学会2011年度年会
  • Place of Presentation: 同志社大学今出川キャンパス
  • Year and Date: 2011-09-16
• [Presentation] On the interpolation constants over triangular elements (2011)
  • Author(s): Kenta Kobayashi
  • Organizer: The 7th East Asia SIAM Conference
  • Place of Presentation: Waseda University Kitakyushu Campus
  • Year and Date: 2011-06-29
• [Presentation] On the interpolation constants over triangular elements (2011)
  • Author(s): Kenta Kobayashi
  • Organizer: The 7th East Asia SIAM Conference
  • Place of Presentation: Waseda University, Kitakyushu Campus
  • Year and Date: 2011-06-29
• [Presentation] 三角形要素上の補間誤差定数について (2010)
  • Author(s): 小林健太
  • Organizer: 研究集会:科学技術計算アルゴリズムの数理的基盤と展開
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-10-19
• [Presentation] 三角形要素上の補間誤差定数について (2010)
  • Author(s): 小林健太
  • Organizer: 研究集会:科学技術計算アルゴリズムの数理的基盤と展開
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Year and Date: 2010-10-19
• [Presentation] On the interpolation constants over triangular elements (2010)
  • Author(s): Kenta Kobayashi
  • Organizer: Conference on Inequalities and Applications '10
  • Place of Presentation: Hajduszoboszlo, Hungary
  • Year and Date: 2010-09-23
• [Presentation] 三角形要素上の補間誤差定 数について (2010)
  • Author(s): 小林健太
  • Organizer: 応用数理学会2010年度年会
  • Place of Presentation: 明治大学駿河台キャンパス
  • Year and Date: 2010-09-08
• [Presentation] 三角形要素上の補間誤差定数について (2010)
  • Author(s): 小林健太
  • Organizer: 応用数理学会2010年度年会
  • Place of Presentation: 明治大学 駿河台キャンパス(東京都)
  • Year and Date: 2010-09-08
• [Presentation] On the interpolation constants over triangular elements (2010)
  • Author(s): Kenta Kobayashi
  • Organizer: Czech-Japanese Seminar in Applied Mathematics
  • Place of Presentation: Czech Technical University, Czech republic
  • Year and Date: 2010-08-31
• [Presentation] On the interpolation constants over triangular elements (2010)
  • Author(s): Kenta Kobayashi
  • Organizer: Czech-Japanese Seminar in Applied Mathematics
  • Place of Presentation: Czech Technical University
  • Year and Date: 2010-08-31
• [Presentation] On the Interpolation Constants over Triangular Elements
  • Author(s): Kenta Kobayashi
  • Organizer: 2012 International Symposium on Nonlinear Theory and its Applications(NOLTA2012)
  • Place of Presentation: Palma, Majorca, Spain