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.

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^^ - ）。该属性称为“超级范围”。通过该项目，我们获得了以下结果：（i）连接 - 扩散问题的隐式有限差方案U_T + div {-k（x，x，y）▽ + + ua + ua} = f（x，x，y）在ωx（0，t）中的差异（ii）差异（ii）差异（ii）numer numers（ii）。在ω是正方形，磁盘或扇区的情况下，拉伸功能已经获得了某些收敛定理。（iii）用于使用绿色功能和离散绿色功能之间的谐波关系的有限差异和有限元元素方法的精确误差分析，我们获得了超级康涅斯（SuperConconvercence）的一些有趣的结果。

项目成果

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: "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。

Q.Fang: "Convergence of inconsistent finite difference schemes for Dirichlet problem whose solution has singular derivatives at the boundary"Information. 4・2. 161-170 (2001)

Q.Fang：“解在边界处具有奇异导数的狄利克雷问题的不一致有限差分格式的收敛性”161-170（2001）。