本文選題：有限元方法 + 梯度重構(gòu)�。� 參考：《南京大學(xué)》2017年碩士論文

【摘要】：本文研究橢圓問題有限元離散的梯度重構(gòu)技術(shù)及其超收斂性質(zhì)。由于基于保多項(xiàng)式梯度重構(gòu)技術(shù)(PPR)的后驗(yàn)誤差估計(jì)在自適應(yīng)有限元方法中的應(yīng)用,其超收斂性質(zhì)受到了人們的青睞與關(guān)注。經(jīng)典的超收斂結(jié)果一般考慮的是Dirichlet邊值問題,本論文考慮Robin邊值問題,并分析在輕度結(jié)構(gòu)化網(wǎng)格下PPR重構(gòu)算子的超收斂性質(zhì)。本論文對(duì)帶Robin邊界條件的二階橢圓問題的線性有限元離散,在三角剖分是輕度結(jié)構(gòu)化的假設(shè)下,證明了有限元解uh的PPR梯度重構(gòu)Ghuh滿足超收斂估計(jì)‖Ghuh-%絬‖L2(Ω)= O(h1+ρ + h2|lnh|1/2),其中0ρ≤1與網(wǎng)格的結(jié)構(gòu)化程度有關(guān)。另外,我們還對(duì)比了幾種邊界點(diǎn)梯度重構(gòu)的方法,發(fā)現(xiàn)它們的超收斂效率相差不大。我們還給出數(shù)值例子驗(yàn)證了理論結(jié)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，

本文編號(hào)：1885239