Pointwise A Posteriori Error Estimators for Multiple and Clustered Eigenvalue Computations

TL;DR

This work develops a pointwise a posteriori error estimator in the $L^{\infty}$-norm for conforming finite element approximations of eigenfunctions corresponding to multiple and clustered eigenvalues of elliptic operators. Key innovations include a Ritz-based framework with the operator $\Lambda_l$, a priori estimates for regularized Green's functions, and a reliability/efficiency analysis that remains robust to eigenvalue gaps and mesh refinements, extended to higher-order elements via weighted Sobolev stability of the $L^2$-projection. The estimator uses both a computable local indicator $\eta_l(T)$ and a theoretical cluster-aware bound $\eta_{l,j}^*$, and an adaptive loop guides refinement to resolve singularities effectively. Numerical experiments on L-shaped and slit-domain geometries corroborate the theory, showing quasi-optimal convergence and edge-residual domination for linear elements, with robust performance across eigenvalue clusters.

Abstract

In this work, we propose an a pointwise a posteriori error estimator for conforming finite element approximations of eigenfunctions corresponding to multiple and clustered eigenvalues of elliptic operators. It is proven that the pointwise a posteriori error estimator is reliable and efficient, up to some logarithmic factors of the mesh size. The constants involved in the reliability and efficiency are independent of the gaps among the targeted eigenvalues, the mesh size and the number of mesh level. Specially, we obtain a by-product that edge residuals dominate the a posteriori error in the sense of $L^{\infty}$-norm when the linear element is used. With the aid of the weighted Sobolev stability of the $L^2$-projection, we also propose a new method to prove the reliability of the a posteriori error estimator for higher order finite elements. A key ingredient in the a posteriori error analysis is some new estimates for regularized derivative Green's functions. Some numerical experiments verify our theoretical results.

Figures (14)

• Figure 1: The adaptive mesh with $19700$ elements in $\Omega_{1}$ for the 12th and 13th eigenpairs.
• Figure 2: The blue dots represent the pointwise a posteriori error estimator $\eta_l$ for $J=\{12,13\}$ in $\Omega_1$. The dotted line represents the fitted convergence rate.
• Figure 3: The convergence rates of two error estimators when using the $H^{1}$ error estimator.
• Figure 4: The convergence rates of two error estimators when using the $L^{\infty}$ error estimator.
• Figure 5: The adaptive mesh with $5373$ elements in $\Omega_{2}$ for the 2nd and 3rd eigenpairs.

Theorems & Definitions (26)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3: Lower bound
• proof
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Remark 4.1