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Construction of Numerical Analysis for High-performance Large-Scale Computation

Research Project

Project/Area Number 13304007
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKyushu University

Principal Investigator

TABATA Masahisa  Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (30093272)

Co-Investigator(Kenkyū-buntansha) KANAYAMA Hiroshi  Kyushu University, Faculty of Engineering, Professor, 大学院・工学研究院, 教授 (90294884)
USHIJIMA Teruo  Univ.of Electro-Communications, Professor, 電気通信学部, 教授 (10012410)
IMAI Hitoshi  Tokushima Univ., Faculty of Engineering, Professor, 工学部, 教授 (80203298)
NAKAO Mitsuhiro  Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (10136418)
KIKUCHI Fumio  Univ.of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40013734)
友枝 謙二  大阪工業大学, 工学部, 教授 (60033916)
大塚 厚二  広島国際学院大学, 工学部, 教授 (30141683)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥40,820,000 (Direct Cost: ¥31,400,000、Indirect Cost: ¥9,420,000)
Fiscal Year 2003: ¥12,480,000 (Direct Cost: ¥9,600,000、Indirect Cost: ¥2,880,000)
Fiscal Year 2002: ¥12,610,000 (Direct Cost: ¥9,700,000、Indirect Cost: ¥2,910,000)
Fiscal Year 2001: ¥15,730,000 (Direct Cost: ¥12,100,000、Indirect Cost: ¥3,630,000)
Keywordsfinite element method / error estimate / numerical simulation / parallel computation / heat convection problems / moving boundary problems / validated computation / Nedelec's edge elements / 温度依存粘性 / 無限精度数値計算 / ヘルムホルツ方程式 / 板曲げ要素 / 非線形拡散 / 特性有限要素法 / 特異極限法 / 精度保証付き数値計算法 / 多孔質媒体流に対する差分法 / ライスナー・ミンドリン型要素 / 自由境界 / 円外帰着波動問題の基本解近似解法 / 一様可解性 / 地球マントル対流
Research Abstract

1.In devising numerical schemes for flow problems, how to approximate the convection torn is a crucial point. Characteristic finite element approximation is based on the approximation of the material derivative, which is the sum of the time derivative term and the convection term. So far finite element schemes of characteristic method of the first-order accuracy in time increment have been used. We have developed a finite element scheme of the second-order accuracy in time increment and obtained the best possible error estimate. This scheme is more robust than the first-order scheme with respect to numerical integration error and can solve flow problems more stably and accurately.
2.We have developed a finite element scheme and established an error estimate for heat convection problems with temperature-dependent viscosity. The viscosity of heat conduction problems such as mantle convection in the Earth and melting glass convection in the furnace is strongly dependent on the temperature. … More The dependence plays an important role in die formation of convection patterns. Our scheme is applicable for the general Rayleigh-Benard problems with temperature-dependent viscosity, thermal conductivity, and thermal expansion coefficient. Using this scheme we have carried out large-scale numerical simulation of Earth's mantle convection in three-dimensional spherical shell and succeeded in obtaining complex heat convection patterns.
3.In the infinite precision computation we have succeeded a large-scale parallel computation using a cluster of high-performance computers with 10CPU and 20GB memory. For one-dimensional boundary-value problems very precise results with precision 4995 digits have been obtained. We have used this system to perform direct numerical simulation of inverse problems and made possible a numerical analysis of inverse problems.
4.Formulating eddy current problems in magnetic vector potential and electric scalar potential, we have solved them using a hierarchical domain decomposition method. This solution has been shown to be effective under the environment of parallel computation. By this method we have carried out large-scale numerical simulation of nonlinear static magnetic problems in magnetic vector potential. Less

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (42 results)

All Other

All Publications (42 results)

  • [Publications] Tabata, M.: "Uniform solvability of finite element solutions in approximate domains"Japan Journal of Industrial and Applied Mathematics. 18. 567-585 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Rui, H., Tabata, M.: "A second order characteristic finite element scheme for convection-diffusion problems"Numerische Mathematik. 92. 161-177 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Tabata, M.: "Finite element approximation to infinite Prandtl number Boussinesq equations with temperature dependent coefficients"Future Generation Computer Systems. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takeuchi, T., Imai, H.: "Direct numerical simulations of Cauchy problems for the Laplace operators"Advances in Mathematical Sciences and Applications. 13. 587-609 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Ushijima, T., Chiba, F.: "A fundamental solution method for the reduced wave problem in a domain exterior to a disc"Journal of Computational and Applied Mathematics. 152. 545-557 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Ohtsuka, K.: "Comparison of criteria on the direction of crack extension"Journal of Computational and Applied Mathematics. 149. 335-339 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kanayama, H., Tagami, D., Imoto, K., Sugimoto, S.: "Finite element computation of magnetic field problems with the displacement current"Journal of Computational and Applied Mathematics. 159. 77-84 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kikuchi, F.: "Theoretical analysis of Nedelec's edge elements"Japan Journal of Industrial and Applied Mathematics. 18. 321-333 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kimura, M., Notsu, H.: "A level set method using the signed distance function"Japan Journal of Industrial and Applied Mathematics. 19. 415-446 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Nakaki, T., Tomoeda, K.: "A finite difference scheme for some nonlinear diffusion equations in absorbing medium : support splitting phenomena"SIAM Journal on Numerical Analysis. 40. 945-964 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Watanabe, Y., Yamamoto, N., Nakao, M.T., Nishida, T.: "A numerical verification of nontrivial solutions for the heat convection problem"Journal of Mathematical Fluid Mechanics. 6. 1-20 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Murakawa, H., Nakaki, T.: "A singular limit approach to moving boundary problems and its applications"Theoretical and Applied Mechanics Japan. 52. 255-260 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Tabata, M.: "Uniform solvability of finite element solutions in approximate domains"Japan Journal of Industrial and Mathematics. 18. 567-585 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Rui, H., Tabata, M.: "A second order characteristic finite element scheme for convection-diffusion problems"Numerische Mathematik. 92. 161-177 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Tabata, M.: "Finite element approximation to infinite Prandtl number Boussinesq equations with temperature dependent coefficients"Future Generation Computer Systems. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takeuchi, T., Imai, H.: "Direct numerical simulations of Cauchy problems for the Laplace operators"Advances in Mathematical Sciences and Applications. 13. 587-609 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Ushijima, T., Chiba, F.: "A fundamental solution method for the reduced wave problem in a domain exterior to a disc"Journal of Computational and Applied Mathematics. 152. 545-557 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Ohtsuka, K.: "Comparison of criteria on the direction of crack extension"Journal of Computational and Applied Mathematics. 149. 335-339 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kanayama, H., Tagami, D., Imoto, K., Sugimoto, S.: "Finite Element Computation of Magnetic Field Problems with the Displacement Current"Journal of Computational and Applied Mathematics. 159. 77-84 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kikuchi, F.: "Theoretical analysis of Nedelecs edge elements"Japan Journal of Industrial and Mathematics. 18. 321-333 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kimura, M., Notsu, H.: "A level set method using the signed distance function"Japan Journal of Industrial and Mathematics. 19. 415-446 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Nakaki, T, Tomoeda, K.: "A finite difference scheme for some nonlinear diffusion equations in absorbing medium : support splitting phenomena"SIAM Journal on Numerical Analysis. 40. 945-964 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Watanabe, Y, Yamamoto, N, Nakao, M.T., Nishida, T.: "A numerical verification of nontrivial solutions for the heat convection problem"Journal of Mathematical Fluid Mechanics. 6. 1-20 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Murakawa, H., Nakaki, T.: "A singular limit approach to moving boundary problems and its applications"Theoretical and Applied Mechanics Japan. 52. 255-260 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Tabata: "Finite element approximation to infinite Prandtl number Boussinessq equations with temperature dependent coefficients"Future Generation Computer Systems. (発表予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Takeuchi, H.Imai: "Direct numerical stimulations of Cauchy problems for the Laplace operators"Advances in Mathematical Sciences and Applications. 13. 587-609 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.ushijima, F.Chiba: "Error estimates for a fundamental solution method applied to reduced wave problems in a domain exterior to a disc"Journal of Computational and Applied Mathematics. 159. 137-148 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] F.Kikuchi, K.Ishii, H.Takahashi: "Ressiner-Mindin extensions of Kirchhoff elements for plate bonding"International journal of Computational Methods. (発表予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] K.tomoeda: "The behavior of impulsively initiated thermal waves in an absorbing medium"Dynamics of Continuous, Discrete and Impulsive Systems Series B : Applications and Algorithm. 10. 151-164 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.T.Nakano, Y.Watanabe, N.Yamamoto, T.Nishida: "Some computer assisted proofs for solutions of the heat convection problems"Reliable Computing. 9. 359-372 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Rui, H., Tabata, M.: "A second order characteristic finite element scheme for convection-diffusion problems"Numerische Mathematik. 92. 161-177 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Nakaki, T., Tomoeda, K.: "A finite difference scheme for some nonlinear diffusion equations in an absorbing medium : support splitting phenomena"SIAM J. Numer. Anal.. 40. 945-964 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kimura, I.M., Notsu, H.: "A level set method using the signed distance function"Japan J. Indust. Appl. Math.. 19. 415-446 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Nagatou, K. et al.: "Verified numerical computations for eigenvalues of non-commutative harmonic oscillators"Numerical Functional Analysis and Optimization. 23. 633-650 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Ohtsuka, K.: "Comparison of criteria on the direction of crack extension"Journal of Computational and Applied Mathematics. 149. 335-339 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Matsuoka, K., Tomoeda, K.: "Numerical computations to pulse splitting phenomena in the thermal waves through an absorbing medium"Journal of Computational and Applied Mathematics. 152. 319-332 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Tabata, M.: "Uniform solvability of finite element solutions in approximate domains"Japan Journal of Industrial and Applied Mathematics. 18. 567-585 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Nakao, M.T.et al.: "Verified numerical computations for an inverse elliptic eigenvalue problem with finite data"Japan Journal of Industrial and Applied Mathematics. 18. 587-602 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Imai, H.et al.: "Numerical computation of Lyapunov exponents related to attractors in a free boundary problem"Nonlinear Analysis. 47. 3823-3833 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kikuchi, F.: "Theoretical analysis of Nedelec's edge elements"Japan Journal of Industrial and Applied Mathematics. 18. 321-333 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Nakaki, T. et al.: "A finite difference scheme for some nonlinear diffusion equations in absorbing medium : support splitting phenomena"SIAM Numerical Analysis. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] Ushijima, T.: "An FEM-CSM combined method for planar exterior Laplace problems"Japan Journal of Industrial and Applied Mathematics. 18. 359-382 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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