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

TL;DR

Addressing large-scale Laplace eigenvalue problems discretized by the nonconforming CR element, the paper develops enhanced error estimates for augmented subspace methods and proves a second-order convergence rate between augmentation steps using spectral projections. It provides explicit CR-based error bounds for single and multiple eigenpairs and sharper algebraic error estimates that reveal dependence on the coarse-space quantity $\eta_a(W_H)$ and eigenvalue gaps $\delta_{k,i}$, with numerical experiments validating the theory. The results improve understanding of coarse-space design and parallelization potential in multilevel eigensolvers employing nonconforming elements, and demonstrate competitive efficiency compared with single-level Krylov-Schur approaches. Overall, the work advances rigorous error control and practical efficiency for augmented subspace eigensolvers in nonconforming finite element settings.

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.

Figures (5)

• Figure 1: Errors for the eigenpair approximations by our algorithms and the single level solver for the first $4$ eigenvalues $2\pi^2$, $5\pi^2$, $5\pi^2$, $8\pi^2$ and their corresponding eigenfunctions with $H=\sqrt{2}/8$.
• Figure 2: CPU time for our algorithms and the single level solver with $H=\sqrt{2}/32$: The left subfigure shows the CPU time for the first eigenpair approximation and the right subfigure shows the CPU time for the smallest $4$ eigenpair approximations.
• Figure 3: The convergence behaviors for the first eigenfunction by Algorithm \ref{['alg:augk_NC']} corresponding to the coarse mesh size $H=\sqrt{2}/8$, $\sqrt{2}/16$, $\sqrt{2}/32$ and $\sqrt{2}/64$, respectively.
• Figure 4: The convergence behaviors for the smallest $4$ eigenfunctions by Algorithm \ref{['alg:augk_NC']} with the coarse space being the linear finite element space on the mesh with size $H=\sqrt{2}/8$, $\sqrt{2}/16$, $\sqrt{2}/32$ and $\sqrt{2}/64$, respectively.
• Figure 5: The convergence behaviors for the only $4$-th eigenfunction by Algorithm \ref{['Algorithm_1']} with the coarse space being the linear finite element space on the mesh with size $H=\sqrt{2}/8$, $\sqrt{2}/16$, $\sqrt{2}/32$ and $\sqrt{2}/64$, respectively.

Theorems & Definitions (16)

• Lemma 2.1: MR2373954lin2012asymptotic
• Lemma 2.2
• proof
• Theorem 2.1
• proof
• Corollary 2.1
• Theorem 2.2
• proof
• Corollary 2.2
• Theorem 3.1