【摘要】：Steklov特征值問題的特征值參數(shù)在邊界條件上,有很強(qiáng)的物理背景.因此,其數(shù)值方法逐步成為學(xué)者們關(guān)注的焦點(diǎn).在偏微分方程的數(shù)值逼近中,基于后驗(yàn)誤差估計(jì)的自適應(yīng)算法因具有計(jì)算量小、計(jì)算時(shí)間短的特點(diǎn),成為有限元方法的主流方向,得到極大的重視.結(jié)合有限元方法及固定位移反迭代,本文提出了Steklov特征值問題的一種基于固定位移反迭代的多網(wǎng)格離散方案.通過該方案,將Steklov特征值問題的解歸結(jié)為首先在粗網(wǎng)格V_H上求特征值問題的解,然后在越來越細(xì)的網(wǎng)格V_(h_i)上求一系列線性代數(shù)方程組的解.本文進(jìn)一步研究了先驗(yàn)誤差估計(jì)和殘差型后驗(yàn)誤差估計(jì),并證明了后驗(yàn)誤差指示子的全局可靠性和局部有效性.此外,基于后驗(yàn)誤差估計(jì),我們設(shè)計(jì)了一種新的固定位移反迭代型的自適應(yīng)算法.這種算法不僅計(jì)算量小而且避免了求解幾乎奇異代數(shù)方程的困難,是一種更為有效的算法.最后,對比三種不同類型的自適應(yīng)算法,用MATLAB編程分別在方形區(qū)域、L-型區(qū)域和菱形裂縫區(qū)域上給出數(shù)值結(jié)果來驗(yàn)證我們方法的有效性.
[Abstract]:The eigenvalue parameters of Steklov eigenvalue problem have a strong physical background on the boundary conditions. Therefore, its numerical method has gradually become the focus of attention of scholars. In the numerical approximation of partial differential equations, the adaptive algorithm based on posterior error estimation has become the mainstream direction of finite element method because of its small amount of calculation and short calculation time, and has been paid great attention to. Based on the finite element method and the inverse iteration of fixed displacement, a multi-grid discretization scheme based on fixed displacement inverse iteration for Steklov eigenvalue problem is proposed in this paper. Through this scheme, the solution of the Steklov eigenvalue problem is reduced to the solution of the eigenvalue problem on the rough grid V 鈮，

本文編號(hào)：2476533