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Development of the hybridized discontinuous method and its mathematical analysis

Research Project

Project/Area Number 26800089
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionWaseda University

Principal Investigator

Oikawa Issei  早稲田大学, 理工学術院, 次席研究員(研究院助教) (10637466)

Project Period (FY) 2014-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2015: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords有限要素法 / 不連続ガレルキン法 / HDG法 / 数値解析 / 不連続Galerkin法 / ハイブリッド法
Outline of Final Research Achievements

We developed and analyzed reduced-order hybridized discontinuous Galerkin (HDG) methods for Poission's equation and the Stokes equations. The proposed method is obtained by introducing the L2-orthogonal projection onto the finite element space of a numerical trace in the stabilization term of the standard HDG method. We mathematically proved that the order of convergence of the proposed method is optimal if we use a polygonal or polyhedral mesh satisfying the so-called chunkiness condition. For the mixed-type HDG method, we devised a new reduced-order scheme by introducing the L2-orthogonal projection in all the discretized equations. It was verified by numerical experiments that the new scheme achieves the optimal-order convergence, however, its error analysis is not provided.

Report

(4 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (15 results)

All 2017 2016 2015 2014

All Journal Article (6 results) (of which Open Access: 2 results,  Acknowledgement Compliant: 6 results,  Peer Reviewed: 4 results) Presentation (9 results) (of which Int'l Joint Research: 3 results,  Invited: 4 results)

  • [Journal Article] HDG methods for second-order elliptic problems2017

    • Author(s)
      Issei Oikawa
    • Journal Title

      RIMS Kokyuroku

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Open Access / Acknowledgement Compliant
  • [Journal Article] A penalty method for the time-dependent Stokes problem with the slip boundary condition and its finite element application2017

    • Author(s)
      Guanyu Zhou, Takahito Kashiwabara and Issei Oikawa
    • Journal Title

      RIMS Kokyuroku

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Open Access / Acknowledgement Compliant
  • [Journal Article] Penalty method with P1/P1 finite element approximation for the Stokes equations under the slip boundary condition2016

    • Author(s)
      Takahito Kashiwabara, Issei Oikawa and Guanyu Zhou
    • Journal Title

      Numerische Mathematik

      Volume: -- Issue: 4 Pages: 705-740

    • DOI

      10.1007/s00211-016-0790-5

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Analysis of a Reduced-Order HDG Method for the Stokes Equations2015

    • Author(s)
      Issei Oikawa
    • Journal Title

      Journal of Scientific Computing

      Volume: -- Issue: 2 Pages: 475-492

    • DOI

      10.1007/s10915-015-0090-8

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Penalty Method for the Stationary Navier–Stokes Problems Under the Slip Boundary Condition2015

    • Author(s)
      Guanyu Zhou, Takahito Kashiwabara and Issei Oikawa
    • Journal Title

      Journal of Scientific Computing

      Volume: -- Issue: 1 Pages: 339-374

    • DOI

      10.1007/s10915-015-0142-0

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A hybridized discontinuous Galerkin method with reduced stabilization2015

    • Author(s)
      Issei Oikawa
    • Journal Title

      Journal of Scientific Computing

      Volume: -- Issue: 1 Pages: 327-340

    • DOI

      10.1007/s10915-014-9962-6

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Reduced stabilization for the HDG method2017

    • Author(s)
      Issei Oikawa
    • Organizer
      Recent Advances in Finite Element Methods
    • Place of Presentation
      City University of Hong Kong, China
    • Year and Date
      2017-03-13
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 実行列に対する実および複素作用素ノルムの違いについて2017

    • Author(s)
      及川一誠
    • Organizer
      研究集会「非線形現象と高精度高品質数値解析」
    • Place of Presentation
      富山大学人間発達科学部
    • Year and Date
      2017-02-13
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] A finite element method for the Stokes equations with a penalized slip boundary condition2016

    • Author(s)
      Issei Oikawa
    • Organizer
      International Workshop on the Multi-Phase Flow; Analysis, Modeling and Numerics
    • Place of Presentation
      Waseda University, Japan
    • Year and Date
      2016-11-10
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] HDG methods for second-order elliptic problems2016

    • Author(s)
      Issei Oikawa
    • Organizer
      RIMS 研究集会「現象解明に向けた数解析学の新展開 II」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2016-10-19
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] HDG methods with reduced stabilization2015

    • Author(s)
      及川一誠
    • Organizer
      RIMS 研究集会「現象解析に向けた数値解析学の新展開」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2015-11-20
    • Related Report
      2015 Research-status Report
  • [Presentation] HDG methods with reduced stabilization2015

    • Author(s)
      Issei Oikawa
    • Organizer
      International Workshop on the Multi-Phase Flow; Analysis, Modeling and Numerics
    • Place of Presentation
      Waseda University
    • Year and Date
      2015-11-11
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] Stokes 方程式に対する弱安定化 HDG 法について2015

    • Author(s)
      及川一誠
    • Organizer
      日本応用数理学会2015年度年会
    • Place of Presentation
      金沢大学
    • Year and Date
      2015-09-11
    • Related Report
      2015 Research-status Report
  • [Presentation] A hybridized discontinuous Galerkin method with reduced stabilization2014

    • Author(s)
      及川一誠
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-25 – 2014-09-29
    • Related Report
      2014 Research-status Report
  • [Presentation] ハイブリッド型不連続Galerkin法の弱安定化スキームについて2014

    • Author(s)
      及川一誠
    • Organizer
      日本応用数理学会年度年会
    • Place of Presentation
      政策研究大学院大学
    • Year and Date
      2014-09-03 – 2014-09-05
    • Related Report
      2014 Research-status Report

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Published: 2014-04-04   Modified: 2018-03-22  

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