Mathematics – Numerical Analysis
Scientific paper
2011-12-06
Mathematics
Numerical Analysis
41 pages
Scientific paper
Finding eigenvalues of partial differential operators is important in the mathematical science. Since the exact eigenvalues are almost impossible, many papers and books investigate their bounds from above and below. It is well known that the variational principle (including the conforming finite element methods) provides the upper bounds, while there are no general theories to provide the lower bounds. The aim of our paper is to introduce a new systematic method that can produce the lower bounds for eigenvalues. The main idea is to use the nonconforming finite element methods. However, the numerics from the literature demonstrate that some nonconforming elements produce upper bounds of eigenvalues though some other nonconforming elements yield lower bounds. The general herein conclusion is that if the local approximation property of the nonconforming finite element space $V_h$ is better than the global continuity property of $V_h$, the corresponding method of the eigenvalue problem will produce the lower bound for the eigenvalue. More precisely, under three hypothesis on the continuity and approximation properties of the nonconforming finite element spaces we first show the abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose a condition and prove that it is sufficient to guarantee the nonconforming finite element methods to produce the lower bounds for eigenvalues of the symmetric elliptic operators. We check the most used nonconforming elements Finally, we prove the saturation condition for most of the nonconforming elements aforementioned.
Hu Jun
Huang Yunqing
Lin Qun
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