【摘要】：自適應(yīng)有限元法是求解橢圓特征值問題的高效數(shù)值方法之一,而后驗誤差估計則是自適應(yīng)有限元方法的理論基礎(chǔ)。1978年美國數(shù)學(xué)家Babuska和Rheinboldt提出有限元后驗誤差估計和自適應(yīng)有限元法的思想。繼他們之后,人們從理論上對有限元自適應(yīng)方法做了大量廣泛的工作,并成功的運(yùn)用到實際應(yīng)用中。結(jié)合協(xié)調(diào)元和自適應(yīng)方法求解橢圓特征值問題,前人做了大量的研究,并得出這個方法的收斂性和優(yōu)越性。運(yùn)用協(xié)調(diào)元和非協(xié)調(diào)元自適應(yīng)方法求解橢圓特征值問題,可以分別得到準(zhǔn)確特征值的上界和下界,這使研究非協(xié)調(diào)元自適應(yīng)方法求解橢圓特征值問題是有意義的。在這樣的背景下,對Laplace特征值問題本文結(jié)合了非協(xié)調(diào)Crouzeix-Raviart元和移位反迭代,首次提出了一種基于殘差型后驗誤差估計的非協(xié)調(diào)Crouzeix-Raviart有限元自適應(yīng)方法。分析了它的收斂性和先驗誤差估計,證明了后驗誤差指示子的有效性和可靠性。最后我們在陳龍的有限元平臺下用MATLAB編程計算,得到了滿意的數(shù)值結(jié)果。
[Abstract]:Adaptive finite element method is one of the efficient numerical methods for solving elliptic eigenvalue problems. The posteriori error estimation is the theoretical basis of adaptive finite element method. In 1978, American mathematicians Babuska and Rheinboldt put forward the idea of finite element posteriori error estimation and adaptive finite element method. After them, people have done a lot of work on the finite element adaptive method in theory, and it has been successfully applied to the practical application. In this paper, a lot of researches have been done to solve the elliptic eigenvalue problem by means of conforming element and adaptive method, and the convergence and superiority of this method have been obtained. The upper and lower bounds of the exact eigenvalues can be obtained by using the conforming element and non-conforming element adaptive methods to solve the elliptic eigenvalue problem respectively, which makes it meaningful to study the non-conforming element adaptive method for solving the elliptic eigenvalue problem. Under this background, a non-conforming Crouzeix-Raviart finite element adaptive method based on residual posteriori error estimation is proposed for the first time in this paper, which combines the nonconforming Crouzeix-Raviart element and the shift inverse iteration for the Laplace eigenvalue problem. Its convergence and prior error estimation are analyzed, and the validity and reliability of the posteriori error indicator are proved. At last, we use MATLAB program to calculate on Chen Long's finite element platform, and obtain satisfactory numerical results.
【學(xué)位授予單位】：貴州師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O241.82

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 ;Eigenvalue approximation from below using non-conforming finite elements[J];Science in China(Series A:Mathematics);2010年01期

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本文編號：2206256