有限体積法によって生成される楕円型作用素の解析半群理論と非線形問題への応用

• Project/Area Number: 18K13460
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Tokyo University of Science
• Principal Investigator: 周 冠宇 東京理科大学, 理学部第一部応用数学科, 助教 (70772705)
• Project Period (FY): 2018-04-01 – 2020-03-31
• Project Status: Discontinued (Fiscal Year 2019)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 有限体積法 / 有限要素法 / Keller--Segel 方程式 / Stokes--Darcy 方程式 / Stokes方程式 / 処罰法 / Keller-Segel方程式 / 離散解析半群

Outline of Annual Research Achievements

(1) 前年度の結果（有限体積法に関する離散半群の理論）を利用して，Keller--Segel 方程式に対する質量と正値性を保存する有限体積法 (FVM) の最適誤差評価を得られた．離散解の事前評価に工夫を入れて，安定性を示した．2017に発表された先行研究の結果に比べて，収束オーダーを改善し，離散解の安定性仮定も不要になった．
意義を重要性：我々はFVMの離散半群理論を複雑な非線形方程式に適用することを成功した．本研究の解析手法は Keller--Segel 方程式だけではなく，他の非線形方程式にも適用できる．FVM 離散半群理論の応用にとって重要な貢献と言える．
(2) DG 要素は離散 Sobolev 空間であり，研究課題に関連する．不連続な Galerkin (DG) 要素を利用する Stokes--Darcy 方程式の Penalty 法と Nitsche's 法の研究を行った．最適な安定性と誤差評価を得るため，我々は DG 要素に関する離散 H^{1/2} ノルムと逆 Trace 作用素の理論を構築した．
意義を重要性：本研究で得られた離散 H^{1/2} ノルムと逆 Trace 作用素に関する理論結果は DG 法の一つの基盤理論であり，様々な問題に応用できる．

Research Products

• [Int'l Joint Research] The Hong Kong Polytechnic University(中国)
• [Int'l Joint Research] The Chinese University of Hong Kong(中国)
• [Int'l Joint Research] The Hong Kong Polytechnic University/The Chinese University of Hong Kong/電子科技大学(中国)
• [Journal Article] Some DG schemes for the Stokes-Darcy problem using P1/P1 element (2019)
  • Author(s): Guanyu Zhou, Takahito Kashiwabara, Issei Oikawa, Eric Chung and Ming-Cheng Shiue
  • Journal Title: Jpn. J. Ind. Appl. Math. Volume: 36
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Penalty method with Crouzeix-Raviart finite element approximation for the Stokes equations under the slip boundary condition (2019)
  • Author(s): Takahito Kashiwabara, Issei Oikawa, and Guanyu Zhou
  • Journal Title: ESAIM: Mathematical Modelling and Numerical Analysis Volume: 印刷中 Issue: 3 Pages: 869-891
  • DOI: 10.1051/m2an/2019008
  • Peer Reviewed
• [Presentation] Keller--Segel 方程式の保存型の有限体積法について (2019)
  • Author(s): 周冠宇
  • Organizer: UTNAS数値解析セミナー
  • Invited
• [Presentation] A penalty method for the Stokes equations with the slip boundary condition on curved boundary (2019)
  • Author(s): Guanyu Zhou
  • Organizer: Workshop for young scholars "Control and inverse problems on waves, oscillations and flows -Mathematical analysis and computational methods-"
  • Int'l Joint Research / Invited
• [Presentation] The penalty approach and Nitsche's method for the interface boundary condition of the Stokes--Darcy problem and the DG approximation (2019)
  • Author(s): Guanyu Zhou
  • Organizer: EASIAM 2019
  • Int'l Joint Research
• [Presentation] On the discrete H^{1/2} norm and the lifting lemmas for the nonconforming elements (2019)
  • Author(s): Guanyu Zhou
  • Organizer: A3 Workshop on uid dynamics and related topics
  • Int'l Joint Research / Invited
• [Presentation] Stokes 方程式の滑り境界問題に対するDG法の解析について (2019)
  • Author(s): 周冠宇
  • Organizer: 不連続 Galerkin 有限要素法の数学理論とその周辺：これからの展開
  • Invited
• [Presentation] Analysis on the fictitious domain method with penalty for the various types of PDEs (2019)
  • Author(s): Guanyu Zhou
  • Organizer: ICIAM2019
  • Int'l Joint Research
• [Presentation] Stokes 方程式の滑り境界問題に対するDG法 の解析について (2019)
  • Author(s): 周 冠宇
  • Organizer: 不連続Galerkin 有限要素法の数学理論とその周辺：これからの展開
• [Presentation] A penalty method to the Stokes-Darcy problem with a smooth interface boundary using the DG element (2018)
  • Author(s): Guanyu Zhou
  • Organizer: EASAIM
  • Int'l Joint Research
• [Presentation] A penalty method for the Stokes-Darcy problem with curved interface boundary and the discontinuous Galerkin approximation (2018)
  • Author(s): Guanyu Zhou
  • Organizer: CSIAM
  • Int'l Joint Research
• [Presentation] 滑らかな界面を持つStokes-Darcy 方程式のDG法について (2018)
  • Author(s): 周 冠宇
  • Organizer: 次世代の科学技術を支える数値解析学の基盤整備と応用展開
  • Invited
• [Presentation] A penalty method to the Stokes-Darcy problem with a smooth interface boundary using the DG element (2018)
  • Author(s): Guanyu Zhou
  • Organizer: The Seventh China-Japan-Korrea Joint Conferrence on Numerrical Mathemattics
  • Int'l Joint Research
• [Funded Workshop] The Seventh China-Japan-Korea Joint Conference on Numerical Mathematics (2018)
• [Funded Workshop] CSIAM (2018)
• [Funded Workshop] EASIAM2018 (2018)