In this paper, we present some enhanced error estimates for augmented subspace methods with the nonconforming Crouzeix-Raviart (CR) element. Before the novel estimates, we derive the explicit error estimates for the case of single eigenpair and multiple eigenpairs based on our defined spectral projection operators, respectively. Then we first strictly prove that the CR element based augmented subspace method exhibits the second-order convergence rate between the two steps of the augmented subspace iteration, which coincides with the practical experimental results. The algebraic error estimates of second order for the augmented subspace method explicitly elucidate the dependence of the convergence rate of the algebraic error on the coarse space, which provides new insights into the performance of the augmented subspace method. Numerical experiments are finally supplied to verify these new estimate results and the efficiency of our algorithms.

翻译：本文针对使用非协调Crouzeix-Raviart（CR）单元的增广子空间方法，提出了一些增强型误差估计。在建立新估计之前，我们首先基于所定义的谱投影算子，分别推导了单特征对和多特征对情形下的显式误差估计。随后，我们首次严格证明了基于CR单元的增广子空间方法在增广子空间迭代的两个步骤之间具有二阶收敛速度，这与实际实验结果相吻合。增广子空间方法的二阶代数误差估计明确揭示了代数误差收敛速度对粗空间的依赖关系，为理解增广子空间方法的性能提供了新的视角。最后通过数值实验验证了这些新的估计结果以及算法的有效性。