Finite difference and finite element analysis for partial differential equations

偏微分方程的有限差分和有限元分析

• 批准号: 11440030
• 负责人: YAMAMOTO Tetsuro
• 金额: $ 4.16万
• 依托单位: Ehime University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440030/
• 关键词: 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 /

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.

本文的出发点是Yamamoto（1998）得到的结果：当u ∈ C^<3，1>（Ω^-）时，边值问题-Δu+f（x，y，u）=0，u=g，在r =δΩ（1）上的S-W有限差分解在x和y方向上的网格尺寸h相等时，在r附近的精度为O（h^3），在其他网格点上的精度为O（h^2）。（1）对流扩散问题u_t + div{-K（x，y）<$u + ua} = f（x，y）在Ω x（0，T）中的隐式差分格式的超收敛性。(ii)在Ω为正方形、圆盘形或扇形的情形下，研究了（1）的不相容差分方法的收敛性及拉伸函数对数值解的加速作用，得到了一些收敛定理。(iii)对两点边值问题的有限差分和有限元方法进行了精细误差分析，利用绿色函数与离散绿色函数之间的调和关系，得到了一些有趣的超收敛结果。

项目成果

N. Matsunaga: "Convergence of Swartztrauber-Sweet's appronimation for the Poisson-type equation on a disk"Numerical Functional Analysis and Optimization. 20. 917-928 (1999)

N. Matsunaga：“盘上泊松型方程的 Swartztrauber-Sweet 近似的收敛性”数值泛函分析和优化。

T. Yamamoto: "Superconvergence and nonsuperconvergence of the Shortley-Weller qppronimations for Dirichlet problems"Numerical Functionla Analysis and Optimization. 22. 455-470 (2001)

T. Yamamoto：“狄利克雷问题的 Shortley-Weller qpronimations 的超收敛和非超收敛”数值函数分析和优化。

K. Yoshida: "Recovered derivatives for the Shortley-Welley finite difference appronimation"Information. 4. 267-277 (2001)

K. Yoshida：“Shortley-Welley 有限差分近似的恢复导数”信息。

Q.Fang: "Superconvergence of finite difference approximations for convection-diffusion problem"Numerical Linear with Applications. 8(印刷中). (2001)

Q.Fang：“对流扩散问题的有限差分近似的超收敛”《数值线性及其应用》（出版中）。

N. Matsunaga: "Comparison of these finite difference appronimations for Dinichlet problems"Information. 2. 55-64 (1999)

N. Matsunaga：“Dinichlet 问题的这些有限差分近似值的比较”信息。