| Project/Area Number |
11440030
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Ehime University |
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Principal Investigator |
YAMAMOTO Tetsuro Ehime Univ., Faculty of Science, Professor, 理学部, 教授 (80034560)
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| Co-Investigator(Kenkyū-buntansha) |
TABATA Masahisa Kyushu Univ., Graduate School of Math., Professor, 大学院・数理学研究院, 教授 (30093272)
FANG Qing Ehime Univ., Faculty of Science, Instructor, 理学部, 助手 (10243544)
TSUCHIYA Takuya Ehime Univ., Faculty of Science, Associate Professor, 理学部, 助教授 (00163832)
CHEN Xiaojun Shimane Univ., Math., Associate Professor, 総合理工学部, 助教授 (70304251)
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| Project Period (FY) |
1999 – 2001
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| Project Status |
Completed (Fiscal Year 2001)
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| Budget Amount *help |
¥6,500,000 (Direct Cost: ¥6,500,000)
Fiscal Year 2001: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2000: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1999: ¥2,700,000 (Direct Cost: ¥2,700,000)
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| Keywords | finite difference method / finite element method / Shortley-Weller qpproximation / superconvergence / boundary value problem / stretching functions / inconsistent scheme / discrete Green's function / Shortley-Weller近似 / 非整合スキームの収束 / 超収束性 / 有限体積法 / 超収束現象 / Swartztrauber-Sweet公式 |
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| Research Abstract |
The starting point of this research is the following result obtained by Yamamoto (1998): The S-W finite difference solution for the boundary value problem
-Δu+f(x,y,u)=0 in Ω, u=g on Γ=δΩ (1)
with equal mesh size h in x and y directions yields O(h^3) accuracy near Γ and O(h^2) accuracy in other grid points, provided that u ∈ C^<3,1>(Ω^^-). This property is called "superconvergence".
Through this project, we obtained the following results:
(i) Superconvergence of the implicit finite difference scheme for the convection-diffusion problem u_t + div{-K(x,y)▽u + ua} = f(x,y) in Ω x (0,T).
(ii) Convergence of inconsistent finite difference methods for (1) and acceleration of the numerical solution by stretching functions in the case where Ω is a square, a disk, or a sector.
Some convergence theorems have been obtained.
(iii) Precise error analysis for finite difference and finite element methods applied to two-point boundary value problems.
By using the harmonic relation between the Green function and the discrete Green function, we obtained some interesting results on superconvergence.
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