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Self-validating numerics with applications to computational science and technology

Research Project

Project/Area Number 10440031
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

KANAO Maitsuhiro  Kyushu University, Graduate school of Mathematics, Proffesor, 大学院・数理学研究科, 教授 (10136418)

Co-Investigator(Kenkyū-buntansha) MUROTA Kazuo  Kyoto University, Res. Inst. Math. Sci., Professor, 数理解析研究所, 教授 (50134466)
NISHIDA Takaaki  Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (70026110)
OISHI shin'ichi  Waseda University, Informatics, Professor, 理工学部, 教授 (20139512)
YAMAMOTO Tetsuro  Ehime University, Mathematics, Professor, 理学部, 教授 (80034560)
MITSUI Taketomo  Nagoya University, Human Informatics, Professor, 大学院・人間情報学研究科, 教授 (50027380)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥11,900,000 (Direct Cost: ¥11,900,000)
Fiscal Year 1999: ¥5,600,000 (Direct Cost: ¥5,600,000)
Fiscal Year 1998: ¥6,300,000 (Direct Cost: ¥6,300,000)
KeywordsNumerical analysis / Validated numerical computation / Numerical verification method / Computer assisted proof
Research Abstract

In this research, we extended and improved the self-validating numerical methods which can be applied to wide mathematical and analytical problems as well as to particular problems such as equations in the mathematical fluid mechanics. The important research results done by investigators and co-investigators are as follows :
1. (by Nakao, N. Yamamoto and Watanabe) Several refinements and extensions were established for the numerical verification methods of solutions for elliptic problems. Namely, the numerical computation with guaranteed error bounds for the eigenvalue problems of second order elliptic operator was established by using the techniques in the numerical verification method of solutions for nonlinear elliptic boundary value problems. We also formulated and obtained basic results for the self-validating method for solutions of elliptic variational inequalities,. Moreover, we presented a verified computation of solutions for the Navier-Stokes equation based on the a posterior … More i and constructive a priori error estimates for the finite element solutions of the Stokes problems. Additionally, we computed a turning point with rigorous error bound for the perturbed and parameterized Gelfand equation.
2. (by Oishi) Some fast algorithms for the fundamental validated computations and the solutions of linear and nonlinear problems were presented.
3. (by Kikuchi) Theoretical and numerical results were obtained for the error analysis of a special kind of finite element method for electro-magnetic problems.
4. (by Sakai) Some applications of splines were presented for plane data approximation.
5. (by Fujino) An efficient acceleration method was investigated for parallel machines.
6. (by Mitsui) A self-validating method for ordinary differential equations with initial value problems was presented.
7. (by T. Yamamoto) Some new error analysis was carried out for the Shortley-Weller type deference scheme for Dirichlet Problems.
8. (by Tabata) Several error estimates were derived of the finite element method for the problem in fluid mechanics.
9. (by Nishida) Some bifurcation phenomena in fluid dynamics were analyzed by the computer assisted proof.
10. (by Murota) The reliability in the structural engineering was investigated by using the group theoretic bifurcation arguments. Less

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (49 results)

All Other

All Publications (49 results)

  • [Publications] Nakao, M.T.: "Constructive L2 error estimates for finite element solutions of the Stokes equations"Reliable Computing. 4. 115-124 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakao, M.T.: "On best constant in the optimal error stimates for the H_< 1> _< 0>-projection into piecewise polynomial spaces"Journal of Approximation Theory. 93. 491-500 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ryoo, C-S: "Numerical verfication of solutions for variational inequalities"Numerische Mathematik. 81. 305-320 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakao, M.T.: "A posteriori constructive a priori error bounds for finite element solutions of Stokes equations"Journal of Computational and Applied Mathematics. 91. 137-158 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakao, M.T.: "Numerical verfications of eigenvalues of second-order elliptic operators"Japan Journal of Industrial and Applied Mathematics. 16. 307-320 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nagatou, K.: "An approach to the numerical verfication of solutions for nonlinear elliptic problems with local uniqueness"Numerical Functional Analysis and Optimization. 20. 543-565 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Watanabe, Y.: "A numerical verfication method of solutions for the Navier-Stokes equations"Reliable Computing. 5. 347-357 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nagatou, K.: "A numerical method of verify the elliptic eigenvalue problems including a uniqueness property"Computing. 63. 109-130 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakao, M.T.: "Numerical verfication of solutions for elasto-plastic torsion problems"to appear in Computers and Mathematics with Applications. 39. 195-204 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kikuchi, F.: "Modification of the 8-node serendipity element"Computer Methods in Applied Mechanics and Engineering. 179. 91-109 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Murota: "Fenchel-type duality for matroid valuations"Mathematical Programming. 82. 357-375 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Tabata: "A precise computation of drag coefficients of a sphere"The International Journal of Computational Fluid Dynamics. 9. 303-311 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Feng, B. -F.: "A conservative spectral method for the third- and fifth-order Korteweg-de Vries equations"Journal of Computational Physics. 153. 467-487 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 中尾充宏: "日本評論社"精度保証付き数値計算. 160 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakao, M. T.: "Constructive LィイD12ィエD1 error estimates for finite element solutions of the Stokes equations,"Reliable Computing. 4. 115-124 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakao, M. T.: "On best constant in the optimal error stimates for the HィイD31(/)0ィエD3-projection into piecewise polynomial spaces"Journal of Approximation Theory. 93. 491-500 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ryoo, C. S.: "Numerical vilification of solutions for variational inequalities"Numerische Mathematik. 81. 305-320 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakao, M. T.: "A posterior and constructive a priori error bounds for finite element solutions of Stokes equations"Journal of Computational and Applied Mathematics. 91. 137-158 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakao, M. T.: "Numerical verifications of eigenvalues of second-order elliptic operators"Japan Journal of Industrial and Applied Mathematics. 16. 307-320 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nagatou, K.: "An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness"Numerical Functional Analysis and Optimization. 20. 543-565 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Watanabe, Y.: "A numerical verification method of solutions for the Navier-Stokes equations"Reliable Computing. 5. 347-357 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nagatou, K.: "A numerical method of verify the elliptic eigenvalue problems including a uniqueness property"Computing. 63. 109-130 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakao, M. T.: "Numerical verification of solutions for elasto-plastic torsion problems"Computers and Mathematics with Applications. 39. 195-204 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kikuchi, F.: "Modification of the 8-node serendipity element"Computer Methods in Applied Mechanics and Engineering. 179. 91-109 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Murota, K.: "Fenchel-type duality for matroid valuations"Mathematical Programming. 82. 357-375 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Tabata, M.: "A precise computation of drag coefficients of a sphere"The International Journal of Computational Fluid Dynamics. 9. 303-311 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Feng, B. F.: "A conservative spectral method for the third- and fifth-order Korteweg-de Vries Equations"Journal of Computational Physics. 153. 467-487 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakao, M. T.: "Validated numerical computations (in Japanese)"Nihon-Hyoron-sha. 160 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakao,M.T.: "Numerical verifications of eigenvalues of second-order elliptic operators"Japan Journal of Industrial and Applied Mathematics. 16. 307-320 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Nagatou,K.: "An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness"Numerical Functional Analysis and Optimization. 20. 543-565 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Watanabe,Y.: "A numerical verification method of solutions for the Navier-Stokes equations"Reliable Computing. 5. 347-357 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Watanabe,Y.: "Verification method of generalized eigenvalue problems and its applications (in Japanese)"Transaction of the Japan Society for Industrial and Applied Mathematics. 9. 137-150 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Nakao,M.T.: "Numerical verification of solutions for elasto-plastic torsion problems"to appear in Computers and Mathematics with Applications.

    • Related Report
      1999 Annual Research Report
  • [Publications] Kikuchi,F.: "Modification of the 8-node serendipity element"Computer Methods in Applied Mechanics and Engineering. 179. 91-109 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Murota,K.: "On the degree of mixed polynomial matrices"SIAM Journal on Matrix Analysis and Applications. 20. 196-227 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Suzuki,A.: "Numerical soluion of an unsteady earth's mantle convection problem by a stabilized finite element method"Theoretical and Applied Mechanics. 48. 371-378 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Feng,B.-F.: "A conservative spectral method for the third-and fofth-order Kortewege-de Vries equations"Journal of Computational Physics. 153. 467-487 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Matsunaga,N.: "Superconvergence of the Shortley-Weller approximation for Dirichlet problems"Journal of Computational and Applied Mathematics. 116(in print). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 中尾充宏: "日本評論社"精度保証付き数値計算. 160 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Nakao,M.T.: "Numerical verification of solutions for nonlinerar elliptic problems using L^∞ residual method" Journal of Mathematical Analysis and Applications. 217. 246-262 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Nakao,M.T.: "On best constant in the optimal error stimates for the H^1_0-projection into piecewise polynomial spaces" Journal of Approximation Theory. 93. 491-500 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Nakao,M.T.: "Constructive L_2 error estimates for finite element solutions of the Stokes equations" Reliable Computing. 4. 115-124 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Nakao,M.T.: "A posteriori and constructive a priori error bounds for finite element solutions of Stokes equations" Journal of Computational and Applied Mathematics. 91. 137-158 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Ryoo,C-S: "Numerical verification of solutions for variational inequalities" Numerische Mathematik. 81. 305-320 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Murota: "Fenchel-type duality for matroid valuations" Mathematical Programming. 82. 357-375 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Tabata: "A precise computation of drag coefficients of a sphere" The International Journal of Computational Fluid Dynamics. 9. 303-311 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Pyi Aye: "Heat Convection of Compressible Fluid" Mathematical Sciences and Applications. 11. 107-115 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] F.Kikuchi: "An improved 4-node quadrilateral plante bending element of the Reissner-Mindlin" Computational Mechanics. 22. 457-471 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 中尾充宏: "日本評論社" 精度保証付き数値計算, 160 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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