The insights gained from the second-order convergence analysis of the augmented subspace method can be leveraged to improve its performance in several ways. One approach could involve adapting the coarse space selection strategy based on the convergence behavior observed in the analysis. By dynamically adjusting the size and properties of the coarse space during the iterations, the method can potentially achieve faster convergence and improved accuracy. Additionally, incorporating adaptive refinement techniques to refine the coarse space in regions where the error is high can help enhance the overall efficiency of the method. Furthermore, exploring alternative interpolation and projection strategies based on the convergence insights can lead to the development of more robust and efficient variants of the augmented subspace method.