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Error analysis of finite element solutions to nonlinear partial differential equations

非线性偏微分方程有限元解的误差分析

基本信息

批准号：
10640123
负责人：
TSUCHIYA Takuya
金额：
$ 1.73万
依托单位：
Ehime University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640123/
关键词：
finite element methods partial differential equations boundary value problems error analysis the inverse function theorem 非線形偏微分方程式

项目摘要

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".

设Ω⊂RィイD1dィエd1是d维欧氏空间Rィイd1dィエd1中的有界域。研究了一类强非线性椭圆型边值问题：∫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)，其中aィイD4→ィエD4，f是充分光滑函数.设F(λ，u)是由上述方程定义的非线性算子。利用Kantorovich定理和带误差估计的隐函数定理，证明了如果(λ，u)是方程的一个精确解，且关于u的Frechet导数DィイD2uィエD2F(λ，u)是某些函数空间之间的同构，则存在一个局部唯一的有限元解(λ，uィイd2hィエD2)接近于(λ，u)，并得到了几个误差估计.这一结果可以通过几种方式进行推广。即使解分支存在转折点，我们也可以得到类似的结果。在这种情况下，有限元解(λィイd2hィエD2，uィイd2hィエD2)的误差估计为|λ-λィイd2hィエD2|+λ-λィイ，我们可以证明，误差|ィエd2hD2|远小于误差||u-uィイd2hD2||。如果方程有对流项，我们必须引入所谓的迎风有限元格式来获得更好的逼近。然而，这种刻画产生了一个不可微的有限元算子。即使如此，如果离散化的算子有“伪导数”，我们也可以得到类似的误差分析。