Pointwise A Posteriori Error Estimators for Multiple and Clustered Eigenvalue Computations

TL;DR

This paper introduces a reliable and efficient pointwise a posteriori error estimator for finite element eigenfunction approximations, effective for multiple and clustered eigenvalues, with proven theoretical guarantees and numerical validation.

Contribution

It develops a novel pointwise a posteriori error estimator that is independent of eigenvalue gaps and mesh size, including new estimates for regularized derivative Green's functions.

Findings

Estimator is reliable and efficient up to logarithmic factors.

Edge residuals dominate the a posteriori error in L-infinity norm for linear elements.

Numerical experiments confirm theoretical results.

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.