带Robin边界条件的椭圆问题的PPR梯度重构

发布时间：2018-05-13 23:12

  本文选题：有限元方法 + 梯度重构　； 参考：《南京大学》2017年硕士论文

【摘要】：本文研究椭圆问题有限元离散的梯度重构技术及其超收敛性质。由于基于保多项式梯度重构技术(PPR)的后验误差估计在自适应有限元方法中的应用,其超收敛性质受到了人们的青睐与关注。经典的超收敛结果一般考虑的是Dirichlet边值问题,本论文考虑Robin边值问题,并分析在轻度结构化网格下PPR重构算子的超收敛性质。本论文对带Robin边界条件的二阶椭圆问题的线性有限元离散,在三角剖分是轻度结构化的假设下,证明了有限元解uh的PPR梯度重构Ghuh满足超收敛估计‖Ghuh-%絬‖L2(Ω)= O(h1+ρ + h2|lnh|1/2),其中0ρ≤1与网格的结构化程度有关。另外,我们还对比了几种边界点梯度重构的方法,发现它们的超收敛效率相差不大。我们还给出数值例子验证了理论结果。
[Abstract]:In this paper, the gradient reconstruction technique for finite element discretization of elliptic problems and its superconvergence property are studied. Due to the application of posteriori error estimation based on preserving polynomial gradient reconstruction technique in adaptive finite element method, its superconvergence property has attracted much attention. The classical superconvergence results generally consider the Dirichlet boundary value problem. In this paper, we consider the Robin boundary value problem and analyze the superconvergence properties of the PPR reconstruction operator on the light structured grid. In this paper, the linear finite element discretization of the second order elliptic problem with Robin boundary condition is given under the assumption that triangulation is mildly structured. It is proved that the PPR gradient reconstruction Ghuh of the finite element solution uh satisfies the superconvergence estimate of Ghuh-% / L _ 2 (惟 _ n = O(h1 蟻 _ h _ 2 lnh _ 1 / 2), where 0 蟻 鈮，

本文编号：1885239