Pointwise A Posteriori Error Estimators for Elliptic Eigenvalue Problems

TL;DR

This paper introduces a new pointwise a posteriori error estimator for elliptic eigenvalue problems, proving its reliability and efficiency in the $L^{ abla}$-norm, and extends the analysis to nonconforming finite element methods with numerical validation.

Contribution

It develops a novel residual-type a posteriori error estimator for elliptic eigenvalues, including a theoretical non-computable estimator and analysis of their relationship, with extensions to nonconforming methods.

Findings

Estimator is reliable and efficient up to a logarithmic factor.

Theoretical and non-computable estimators are analyzed and related.

Numerical experiments confirm the theoretical results.

Abstract

In this work, we propose and analyze a pointwise a posteriori error estimator for simple eigenvalues of elliptic eigenvalue problems with adaptive finite element methods (AFEMs). We prove the reliability and efficiency of the residual-type a posteriori error estimator in the sense of $L^{\infty}$-norm, up to a logarithmic factor of the mesh size. For theoretical analysis, we also propose a theoretical and non-computable estimator, and then analyze the relationship between computable estimator and theoretical estimator. A key ingredient in the a posteriori error analysis is some new estimates for regularized derivative Green's functions. This methodology is also extended to the nonconforming finite element approximations. Some numerical experiments verify our theoretical results.