基于移位反迭代的非协调Crouzeix-Raviart有限元自适应方法求Laplace特征值问题
发布时间：2018-08-26 21:47
【摘要】：自适应有限元法是求解椭圆特征值问题的高效数值方法之一,而后验误差估计则是自适应有限元方法的理论基础。1978年美国数学家Babuska和Rheinboldt提出有限元后验误差估计和自适应有限元法的思想。继他们之后,人们从理论上对有限元自适应方法做了大量广泛的工作,并成功的运用到实际应用中。结合协调元和自适应方法求解椭圆特征值问题,前人做了大量的研究,并得出这个方法的收敛性和优越性。运用协调元和非协调元自适应方法求解椭圆特征值问题,可以分别得到准确特征值的上界和下界,这使研究非协调元自适应方法求解椭圆特征值问题是有意义的。在这样的背景下,对Laplace特征值问题本文结合了非协调Crouzeix-Raviart元和移位反迭代,首次提出了一种基于残差型后验误差估计的非协调Crouzeix-Raviart有限元自适应方法。分析了它的收敛性和先验误差估计,证明了后验误差指示子的有效性和可靠性。最后我们在陈龙的有限元平台下用MATLAB编程计算,得到了满意的数值结果。
[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.
【学位授予单位】：贵州师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82
本文编号：2206256
[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.
【学位授予单位】：贵州师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82
【参考文献】
相关期刊论文 前1条
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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