Pointwise A Posteriori Error Estimators for Elliptic Eigenvalue Problems

TL;DR

This paper develops a pointwise (in $L^{\infty}$) a posteriori error estimator for simple eigenvalues of second-order elliptic eigenvalue problems solved by adaptive finite element methods (AFEM). It introduces a computable estimator $\eta$ and a theoretical estimator $\eta^*$, and proves reliability and efficiency of $\eta$ up to a logarithmic factor in the mesh size, by leveraging new regularized derivative Green's function estimates. A key technical contribution is the $L^1$-type control of the Galerkin approximation to the regularized Green's functions, enabling a Aubin–Nitsche-type argument in this pointwise setting; the framework extends to Crouzeix–Raviart nonconforming elements. Numerical experiments on domains with singularities (e.g., L-shaped and slit domains) verify the theory and show quasi-optimal convergence rates for the estimator, highlighting the practical impact for robust eigenvalue computations with AFEM.

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.

Figures (4)

• Figure 1: The adaptive mesh with $7822$ elements in L-shape domain $(0,\pi)^2 \backslash ([\frac{\pi}{2},\pi)\times (0,\frac{\pi}{2}])$ for the first eigenpair.
• Figure 2: The blue dots represent the pointwise a posteriori error estimator $\eta$ for the first eigenpair in L-shape domain $(0,\pi)^2 \backslash ([\frac{\pi}{2},\pi)\times (0,\frac{\pi}{2}])$. The dotted line represents the fitted convergence rate.
• Figure 3: The adaptive mesh with $9971$ elements in slit domain $\Omega=\{(x,y): \vert x\vert +\vert y\vert <1\}$ with the crack $[0,1]\times \{0\}$ for the first eigenpair.
• Figure 4: The blue dots represent the pointwise a posteriori error estimator $\eta$ for the first eigenpair in slit domain $\Omega=\{(x,y): \vert x\vert +\vert y\vert <1\}$ with the crack $[0,1]\times \{0\}$. The dotted line represents the fitted convergence rate.

Theorems & Definitions (22)

• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7
• proof