Abstract
In this paper, some enhanced error estimates are derived for the augmented subspace methods which are designed for solving eigenvalue problems. For the first time, we strictly prove that the augmented subspace methods have the second order convergence rate between the two iteration steps, which is better than the existing theoretical results in Lin and Xie (Math Comp 84:71–88, 2015), Xie et al. (SIAM J Numer Anal 57(6):2519–2550, 2019), Xu et al. (SIAM J Sci Comput 42(5):A2655–A2677, 2020) and more consistent with the results of actual numerical test. These sharper estimates explicitly depict the dependence of convergence rate on the coarse spaces, which provides new advantages for the augmented subspace methods. Some numerical examples are finally presented to validate these new estimate results and the efficiency of our algorithms.
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