| 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)
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| 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".
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