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1999 Fiscal Year Final Research Report Summary

Error analysis of finite element solutions to nonlinear partial differential equations

Research Project

Project/Area Number 10640123
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

TSUCHIYA Takuya  Faculty of Science, Ehime University, associate professor, 理学部, 助教授 (00163832)

Co-Investigator(Kenkyū-buntansha) HASHIMOTO Takahiro  Faculty of Science, Ehime University, assistant professor, 理学部, 助手 (60291499)
FANG Qing  Faculty of Science, Ehime University, assistant professor, 理学部, 助手 (10243544)
YAAMAMOTO Tetsuro  Faculty of Science, Ehime University, professor, 理学部, 教授 (80034560)
Project Period (FY) 1998 – 1999
Keywordsfinite element methods / partial differential equations / boundary value problems / error analysis / the inverse function theorem
Research Abstract

Let Ω ⊂ RィイD1dィエD1 be a bounded domain in the d-dimensional Euclidean space RィイD1dィエD1. The following strongly nonlinear elliptic boundary value problem has been considered :
∫ィイD2ΩィエD2(aィイD4→ィエD4(λ,x,u,∇u)・∇ν+f(λ,x,u,∇u)ν)=0, ∀ν ∈ ΗィイD31(/)0ィエD3(Ω),
where aィイD4→ィエD4, f are sufficiently smooth functions. Let F(λ,u) be the nonlinear operator defined by the above equation. We have shown, using the Kantorovich theorem and the Implicit Function Theorem with error estimation, that if (λ,u) is an exact solution of the equation and the Frechet derivative DィイD2uィエD2F(λ,u) with respect to u is an isomorphism between certain function spaces then there exists a locally unique finite element solution (λ,uィイD2hィエD2) closed to (λ,u) and several error estimates are obtained. This result can be extended in a few ways. Even if solution branch has turning points we can obtain similar results. In such a case, the error of the finite element solution (λィイD2hィエD2,uィイD2hィエD2) is estimated as
|λ-λィイD2hィエD2|+ ||u-uィイD2hィエD2||<_C||u-ΠィイD2hィエD2u||.
Moreover, we can show that the error |λ-λィイD2hィエD2| is much smaller that the error ||u-uィイD2hィエD2||. If the equation has a convection term, we have to introduce so-called upwind finite element scheme to obtain better approximation. However, such kind of discritization yields a non-differentiable finite element operator. Even so, we can obtain similar error analysis if the discirtized operator has a "pseudo-derivative".

  • Research Products

    (8 results)

All Other

All Publications (8 results)

  • [Publications] T.Tsuchiya: "An application of the Kantorovich theorem to nonlinear finite eleme analysis:"Numerische Mathematik. 84. 121-141 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Tsuchiya: "Finite element analysis for parametrized nonlinear equations around furning points"Journal of comprtational and Applied Mathematics. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] N.Matsunaga,T.Tsuchiya: "Non-differentiable finite element approximations for parametrized strongly nonlinear boundary value problems"Advances in Mathematical Sciencies and Appli. (accepted).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Tsuchiya: "Finite element approximations of parametrized strongly nonlinocer boundary value problem"(投稿中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T. Tsuchiya: "An application of the Kantorovich Theorem to nonlinear finite element analysis"Numerishche Mathematik. 84. 121-141 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Tsuchiya: "Finite element analysis for parametrized nonlinear equations around turning points"Journal of Computational and Applied Mathematics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Nami Matsunaga, T. Tsuchiya: "Non-differentiable finite element approximations for parametrized strongly nonlinear noundary value problems"Advances in Mathematical Sciences and Applications. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Tsuchiya: "Finite element approximations of parametrized strongly nonlinear boundary value problems"(submitted).

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

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Published: 2001-10-23  

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