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2006 Fiscal Year Final Research Report Summary

Synthetic approach for the development of computer assisted analysis from the numerical verification methods

Research Project

Project/Area Number 15204007
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 UNIVERCITY

Principal Investigator

NAKAO Mitsuhiro  Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (10136418)

Co-Investigator(Kenkyū-buntansha) TABATA Masahisa  Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (30093272)
IMAI Hitoshi  Tokushima University, Faculty of Engineering, Professor, 工学部, 教授 (80203298)
TSUCHIYA Takuya  Ehime University, Faculty of Science, Professor, 理学部, 教授 (00163832)
NISHIDA Takaaki  Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (70026110)
CHIN Shokun  Hirosaki University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70304251)
Project Period (FY) 2003 – 2006
KeywordsNumerical analysis / Validated computation / Numerical verification / Computer assisted proof / 計算機援用証明
Research Abstract

In this research, we newly developed the numerical verification methods which can be applied to wide mathematical and analytical problems, as well as extended or improved the existing techniques.
And we actually applied these methods to particular problems such as equations in the mathematical fluid mechanics and oscillation problems etc. The important research results obtained by investigators and co-investigators are as follows :
1.Nakao, N.Yamamoto, Watanabe established several refinements and extensions for the constructive error estimates for the finite finite element projections of the Poisson and the bi-harmonic equations on various kinds of domains, particularly, on nonconvex polygonal domains. These results played important and essential roles for the numerical verification of solutions of nonlinear elliptic equations and the two dimensional stationary Navier-Stokes problems.
2.Nagatou numerically proved the stability of the flow on the torus called Kolmogorov problem.
3.Minamoto presented a formulation of the verification condition for the double turning point and applied it to the perturbed Gelfand equation.
4.Oishi established some refinements on the fast algorithm for the solutions of linear equations.
5.Nishida et al. presented the computed results with guaranteed error bounds for the symmetry breaking bifurcation point of the solution of two dimensional heat convection problems, as well as they formulated the numerical verification algorithm for the three dimensional problems with some prototypical verified examples.
6.Chin obtained some numerical verification results on the existence of solutions and a posteriori error estimates for the linear complementarity problems.

  • Research Products

    (18 results)

All 2007 2006 2005 2004 2003 Other

All Journal Article (18 results)

  • [Journal Article] Numerical verification of stationary solutions for Navier-Stokes problems2007

    • Author(s)
      Nagatou, K.
    • Journal Title

      Journal of Computational and Applied Mathematics 199

      Pages: 445-451

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Numerical method for verifying the existence and local uniqueness of a double turning point for a radically symmetric solution of the perturbed Gelfand equation2007

    • Author(s)
      Minamoto, T.
    • Journal Title

      Journal of Computational and Applied Mathematics 202

      Pages: 177-185

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Discrepancy between theory and real computation on the stability of some finite element schemes2007

    • Author(s)
      M.Tabata
    • Journal Title

      Journal of Computational and Applied Mathematics 199

      Pages: 424-431

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On a linearized system arising in the study of Benard-Marangoni convection2007

    • Author(s)
      T.Nishida
    • Journal Title

      Proc.Navier-Stokes Equations and Applications, New Series of Kokyuroku, RIMS, Kyoto University

      Pages: 1-19

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On error estimation of finite element approximations to the elliptic equations in nonconvex polygonal domains2007

    • Author(s)
      N.Yamamoto
    • Journal Title

      Journal of Computational and Applied Mathematics 199

      Pages: 286-296

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Some computer assisted proofs on three dimensional heat convection problems2006

    • Author(s)
      M.-N.Ki
    • Journal Title

      Proceedings of Nonlinear Theory and its Applications NOLTA 2006,11-14 September, Bologna, Italy

      Pages: 427-430

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Existence of solutions to systems of underdetermined equations and spherical designs2006

    • Author(s)
      X.Chen
    • Journal Title

      SIAM Journal on Numerical Analysis 44

      Pages: 2326-2341

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A practical method of numerical calculation of the mapping degree2006

    • Author(s)
      S.Murashige
    • Journal Title

      IEICE Trans. on Fundamentals of Electronics, Communications and Computer Sciences E89-A

      Pages: 1813-1819

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A numerical method to verify the invertibility of linear elliptic operators with applicationsto nonlinear problems2005

    • Author(s)
      Nakao, M. T.
    • Journal Title

      Computing 75

      Pages: 1-14

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Numerical Verification Methods of Solutions for the Free Boundary Problems2005

    • Author(s)
      Hashimoto, K.
    • Journal Title

      Numerical Functional Analysis and Optimization 26

      Pages: 523-542

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A numerical method to verify the inevitability of linear elliptic operators with applications to nonlinear problems2005

    • Author(s)
      Nakao, M.T.
    • Journal Title

      Computing 75

      Pages: 1-14

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A Numerical Verification of Nontrivial Solutions for the Heat Convection Problem2004

    • Author(s)
      Watanabe, Y.
    • Journal Title

      Journal of Mathematical Fluid Mechanics 6

      Pages: 1-20

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] An efficient approach to the numerical verification for solutions of elliptic differential equations2004

    • Author(s)
      Nakao, M. T.
    • Journal Title

      Numerical Algorithms 37

      Pages: 311-323

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A Numerical Verification of Nontrivial Solutions for the Heat Convection Problem2004

    • Author(s)
      Y.Watanabe
    • Journal Title

      Journal of Mathematical Fluid Mechanics 6

      Pages: 1-20

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] An efficient approach to the numerical verification for solutions of elliptic differential equations2004

    • Author(s)
      Nakao, M.T., Y.
    • Journal Title

      Numerical Algorithms 37, Special issue for Proceedings of SCAN2002

      Pages: 311-323

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Some computer assisted proofs for solutions of the heat convection problems2003

    • Author(s)
      Nakao, M. T.
    • Journal Title

      Reliable Computing 9

      Pages: 359-372

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Some computer assisted proofs for solutions of the heat convection problems2003

    • Author(s)
      Nakao, M.T.
    • Journal Title

      Reliable Computing 9

      Pages: 359-372

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A computational approach to constructive a priori and a posteriori error estimates for finite element approximations of bi-harmonic problems

    • Author(s)
      Nakao, M.T.
    • Journal Title

      GAKUTO International Series, Mathematical Sciences and Applications, Proceedings of the 4th JSIAM-SIMAI Seminar on Industrial and Applied Mathematics, May 26-28,2005,Hayama, Japan (to appear)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2008-05-27  

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