Pointwise A Posteriori Error Estimators for Multiple and Clustered Eigenvalue Computations

2025/11/11 by Z. G. Li, Qigang Liang, Li, Zhenglei +3 · 1 citation
Computer Science · Decision Sciences · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2511.08259

openalex publication_date 2025/11/11 · openalex created_date 2025/11/13 · openalex updated_date 2026/07/28

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^∞-norm when the linear element is used. With the aid of the weighted Sobolev stability of the L²-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.

Citations

• Pointwise A Posteriori Error Estimators for Elliptic Eigenvalue Problems
• Pointwise error estimates of linear finite element method for Neumann boundary value problems in a smooth domain
• Convergence and optimality of higher-order adaptive finite element\n methods for eigenvalue clusters
• Convergence Rate and Quasi-Optimal Complexity of Adaptive Finite Element Computations for Multiple Eigenvalues
• Local energy estimates for the finite element method on sharply varying grids
• A unifying theory of a posteriori finite element error control
• Bounds of Riesz Transforms on L^p Spaces for Second Order Elliptic Operators
• Superconvergence in Finite Element Methods and Meshes That are Locally Symmetric with Respect to a Point

Cited by

• An a Posteriori Error Estimator for C⁰ IPG Approximation of Multiple and Clustered Eigenvalues of the Biharmonic Operator

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• Reliable a-posteriori error estimators for hp-adaptive finite element approximations of eigenvalue/eigenvector problems
• Asymptotically exact a posteriori error estimates of eigenvalues by the Crouzeix-Raviart element and enriched Crouzeix-Raviart element