Enhanced Error Estimates for Augmented Subspace Method with Crouzeix-Raviart Element

TL;DR

This paper develops enhanced second-order error estimates for augmented subspace methods using the Crouzeix-Raviart element, supported by theoretical proofs and numerical experiments demonstrating improved convergence and efficiency.

Contribution

The paper introduces explicit second-order error estimates for augmented subspace methods with CR elements, including proofs and numerical validation of improved convergence rates.

Findings

Second-order convergence rate between augmented subspace iteration steps.

Explicit algebraic error estimates depending on the coarse space.

Numerical experiments confirming theoretical error estimates and algorithm efficiency.

Abstract

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.