Research on high accuracy computing and numerical verification for Finite Element Method solution in a non-convex domain

• Project/Area Number: 19740052
• 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: Kanazawa University
• Principal Investigator: KOBAYSHI Kenta Kanazawa University, 数物科学系, 准教授 (60432902)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,800,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥600,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2007: ¥1,200,000 (Direct Cost: ¥1,200,000)
• Keywords: 精度保証 / 有限要素法 / 誤差評価 / 非凸領域 / ポアソン方程式 / 重調和方程式 / Navier-Stokes方程式 / ボアソン方程式

Research Abstract

In solving partial differential equation by Finite Element Method in a non-convex domain, it is known that the convergent rate could be improved by adding singularity functions to the Finite Element basis or using mesh refinement. In our research, we have obtained explicit error estimations for these problems. These results can be applied for computer-assisted proof for non-linear problems.

Research Products

• [Journal Article] A constructive a priori error estimation for finite element discretizations in a non-convex domain using singular functions (2009)
  • Author(s): K. Kobayashi
  • Journal Title: Japan Journal of Industrial and Applied Mathematics Vol.26[2/3] Pages: 493-516
  • NAID: 10028169365
  • Peer Reviewed
• [Journal Article] On the critical case of Okamoto's continuous non-differentiable functions (2009)
  • Author(s): K. Kobayashi
  • Journal Title: Proceedings of the Japan Academy, Series A Vol.85[8] Pages: 101-104
  • NAID: 120005447707
  • Peer Reviewed
• [Journal Article] On the critical case of Okamoto's continuous non-differentiable functions (2009)
  • Author(s): Kenta Kobayashi
  • Journal Title: Proceedings of the Japan Academy, Series A 85[8] Pages: 101-104
  • NAID: 120005447707
  • Peer Reviewed
• [Journal Article] A constructive a priori error estimation for finite element discretizations in a non-convex domain using singular functions (2009)
  • Author(s): Kenta Kobayashi
  • Journal Title: Japan Journal of Industrial and Applied Mathematics 26[2/3] Pages: 493-516
  • NAID: 10028169365
  • Peer Reviewed
• [Journal Article] A constructive a priori error estimation for finite element discretizations in a non-convex domain using singular functions (2009)
  • Author(s): Kobayashi, K
  • Journal Title: Japan Journal of Industrial and Applied Mathematics Vol.26(掲載決定)
  • NAID: 10028169365
  • Peer Reviewed
• [Journal Article] メッシュリファイメントを用いた有限要素解の事前誤差評価 (2008)
  • Author(s): 小林健太
  • Journal Title: 京都大学数理解析研究所講究録 Vol.1614 Pages: 180-186
• [Journal Article] Verified numerical computation of solutions for the stationary Navier-Stokes equation in nonconvex polygonal domains (2007)
  • Author(s): K. Hashimoto, K. Kobayashi, M.T. Nakao
  • Journal Title: Hokkaido Mathematical Journal vol.36[4] Pages: 777-799
• [Journal Article] Verified numerical computation of solutions for the stationary Navier-Stokes equation in nonconvex polygonal domains (2007)
  • Author(s): Nakao, M. T.; Hashimoto, K.; Kobayasbi, K.
  • Journal Title: Hokkaido Mathematical Journal 36 Pages: 777-799
  • Peer Reviewed
• [Journal Article] On the global uniqueness of Stokes' wave of extreme form
  • Author(s): K. Kobayashi
  • Journal Title: IMA Journal of Applied Mathematics (掲載決定)
  • Peer Reviewed
• [Presentation] On the interpolation constant over triangular elements (2010)
  • Author(s): Kenta Kobayashi
  • Organizer: International Workshop on Numerical Verification and its Applications 2010 (INVA2010)
  • Place of Presentation: Hotel Lido Azzurro(Tokyo)
  • Year and Date: 2010-03-13
• [Presentation] On the interpolation constant over triangular elements (2010)
  • Author(s): Kenta Kobayashi
  • Organizer: International Workshop on Numerical Verification and its Applications 2010(INVA2010)
  • Place of Presentation: Hotel Lido Azzurro, Hachijo-jima(東京都)
  • Year and Date: 2010-03-13
• [Presentation] 三角形領域上の補間誤差定数について (2010)
  • Author(s): 小林健太
  • Organizer: 北陸応用数理研究会2010
  • Place of Presentation: 金沢大学サテライト・プラザ(石川県)
  • Year and Date: 2010-02-12
• [Presentation] An a priori constant which appear in bi-harmonic problems (2009)
  • Author(s): Kenta Kobayashi
  • Organizer: Dagstuhl Seminar: Computer-assisted proofs - tools, methods and applications
  • Place of Presentation: Schloss Dagstuhl(Germany)
  • Year and Date: 2009-11-18
• [Presentation] An a priori constant which appear in bi-harmonic problems (2009)
  • Author(s): Kenta Kobayashi
  • Organizer: Dagstuhl Seminar : Computer-assisted proofs-tools, methods and applications
  • Place of Presentation: Schloss Dagstuhl, Wadern,(ドイツ)Germany
  • Year and Date: 2009-11-18
• [Presentation] A constructive a priori error estimation for finite element discretizations in a non-convex domain using mesh refinement (2009)
  • Author(s): Kenta Kobayashi
  • Organizer: International Conference on Engineering and Computational Mathematics (ECM2009)
  • Place of Presentation: The Hong Kong Polytechnic University(China)
  • Year and Date: 2009-05-28
• [Presentation] A constructive a priori error estimation for finite element discretizations in a non-convex domain using mesh refinement (2009)
  • Author(s): Kenta Kobayashi
  • Organizer: International Conference on Engineering and Computational Mathematics(ECM2009)
  • Place of Presentation: The Hong Kong Polytechnic(中国)University
  • Year and Date: 2009-05-28
• [Presentation] An a priori constant which appears in bi-harmonic problems (2009)
  • Author(s): Kenta Kobayashi
  • Organizer: International Workshop on Numerical Verification and its Applications 2009 (INVA2009)
  • Place of Presentation: Hotel Breeze Bay Marina(Okinawa)
  • Year and Date: 2009-03-24
• [Presentation] An a priori constant which appears in bi-harmonic problems (2009)
  • Author(s): Kobayashi, K
  • Organizer: International Workshop on Numerical Verification and its Applications 2009
  • Place of Presentation: Hotel Breeze Bay Marina, Mivako Island, Okinawa
  • Year and Date: 2009-03-24
• [Presentation] Numerical verification method and its applications (2009)
  • Author(s): 小林健太
  • Organizer: 第10回北東数学解析研究会
  • Place of Presentation: 東北大学理学部(宮城県)
  • Year and Date: 2009-02-17
• [Presentation] A constructive a priori error estimation for finite element discretizations in a non-convex domain using a mesh refinement (2009)
  • Author(s): Kenta Kobayashi
  • Organizer: 1st African Conference on Computational Mechanics (AfriCOMP 2009)
  • Place of Presentation: Sun City Convention Center(South Africa)
  • Year and Date: 2009-01-07
• [Presentation] A constructive a priori error estimation for finite element discretizations in a non-convex domain using mesh refinement (2009)
  • Author(s): Kobayashi, K
  • Organizer: 1st African Conference on Computational Mechanics
  • Place of Presentation: Sun City, South Africa
  • Year and Date: 2009-01-07
• [Presentation] 三角形要素上の補間誤差定数について (2008)
  • Author(s): 小林健太
  • Organizer: 流れ問題のための高品質数値解法と計算機援用解析学
  • Place of Presentation: KKRホテル金沢(石川県)
  • Year and Date: 2008-11-18
• [Presentation] 非凸領域における有限要素法の事前誤差評価について (2008)
  • Author(s): 小林健太
  • Organizer: 第3回岐阜非線形ワークショップ-計算科学の基盤としての数値解析-
  • Place of Presentation: 岐阜大学地域科学部棟(岐阜県)
  • Year and Date: 2008-10-27
• [Presentation] A constructive a priori error estimation for finite element discretizations in a non-convex domain using mesh refinement (2008)
  • Author(s): Kenta Kobayashi
  • Organizer: International Workshop on Numerical Verification and its Applications 2008 (INVA2008)
  • Place of Presentation: Tokyo Dai-ichi Hotel OKINAWA GRAND MER RESORT(Okinawa)
  • Year and Date: 2008-03-06
• [Presentation] メッシュリファイメントを用いた有限要素解の事前誤差評価 (2007)
  • Author(s): 小林健太
  • Organizer: 研究集会:計算科学の基盤技術としての高速アルゴリズムとその周辺
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Year and Date: 2007-11-16
• [Presentation] メッシュリファイメントを用いた有限要素解の事前誤差評価 (2007)
  • Author(s): 小林 健太
  • Organizer: RIMS研究集会:計算科学の基盤技術としての高速アルゴリズムとその周辺
  • Place of Presentation: 京都大学 数理解析研究所
  • Year and Date: 2007-11-16
• [Presentation] A constructive a priori error estimation for finite element discretizations in a non-convex domain using mesh refinement (2007)
  • Author(s): Kenta Kobayashi
  • Organizer: The 3rd East Asia SIAM Conference
  • Place of Presentation: Xiamen University(China)
  • Year and Date: 2007-11-03
• [Presentation] A Constructive a Priori Error Estimation for Finite Element Discretizations in a Non-convex Domain Using Mesh Refinement (2007)
  • Author(s): Kobayashi, K.
  • Organizer: The 3rd East Asia SIAM Conference
  • Place of Presentation: Xiamen, China
  • Year and Date: 2007-11-03
• [Presentation] Michael Plum, Some numerical results for the explicit L∞error bounds (2007)
  • Author(s): Kenta Kobayashi
  • Organizer: COE Conference on the Development of Dynamic Mathematics with High Functionality (DMHF2007)
  • Place of Presentation: Fukuoka Recent Hotel(Fukuoka)
  • Year and Date: 2007-10-04
• [Presentation] Some numerical results for the explicit L∞ error bounds (2007)
  • Author(s): Kobayashi, K.
  • Organizer: Conference on the Development of Dynamic Mathematics with High Functionality
  • Place of Presentation: Fukuoka, Japan
  • Year and Date: 2007-10-04
• [Presentation] 水面波方程式に対する計算機援用証明 (2007)
  • Author(s): 小林健太
  • Organizer: 日本数学会2007年度秋季総合分科会
  • Place of Presentation: 東北大学川内北キャンパス(宮城県)
  • Year and Date: 2007-09-23
• [Presentation] A constructive a priori error estimation for finite element discretizations in a non-convex domain using mesh refinement (2007)
  • Author(s): Kenta Kobayashi
  • Organizer: Conference on Inequalities and Applications '07
  • Place of Presentation: De La Motte Castle(Hungary)
  • Year and Date: 2007-09-11
• [Presentation] A Constructive a Priori Error Estimation for Finite Element Discretizations in a Non-convex Domain Using Mesh Refinement (2007)
  • Author(s): Kobayashi, K.
  • Organizer: Conference on Inequalities and Applications '07
  • Place of Presentation: Noszvaj, Hungary
  • Year and Date: 2007-09-11
• [Presentation] A constructive a priori error estimation for finite element discretizations in a non-convex domain using mesh refinement (2007)
  • Author(s): Kenta Kobayashi
  • Organizer: International Conference on Recent Developments of Numerical Schemes for Flow Problems (INSF2007)
  • Place of Presentation: Kyushu University Nishijin Plaza(Fukuoka)
  • Year and Date: 2007-06-29
• [Presentation] A Constructive a Priori Error Estimation for Finite Element Discretizations in a Non-convex Domain Using Mesh Refinement (2007)
  • Author(s): Kobayashi, K.
  • Organizer: International Conference on Recent Developments of Numerical Schemes for Flow Problems
  • Place of Presentation: Kyushu University
  • Year and Date: 2007-06-29